Let L be an integer >= 4 and Lambda = (Z/LZ)^2 the periodic square
torus. Put one spin q_x in S^2 at each site x, with sigma the
normalised area measure on S^2. The Gibbs probability is
dmu_{beta,Lambda}(q)
= (1 / Z_{beta,Lambda})
exp( beta * sum_{{x,y} in E(Lambda)} q_x . q_y )
prod_{x in Lambda} dsigma(q_x), (0.1)
with E(Lambda) the set of nearest-neighbour unoriented edges (each
edge counted once) and Z_{beta,Lambda} the partition function.
beta > 0 is the inverse coupling. There is no external field and
no topological term. Internal O(3) acts on every spin at once.
Each edge {x,y} contributes beta q_x.q_y = beta cos angle(q_x,q_y).
Using |q_x-q_y|^2 = 2 - 2 q_x.q_y,
beta q_x . q_y = -(beta/2) |q_x - q_y|^2 + beta. (0.2)
So a large nearest-neighbour spin difference q_x - q_y costs
kinetic energy; the model prefers neighbouring spins aligned when
beta is large. This second form is what makes "rough bond"
statements (|q_x-q_y| > t for a small threshold t) natural
throughout the proof.
For any beta and any finite L, mu_{beta,Lambda} is well defined.
Standard compactness of S^2 and tightness of the family gives at
least cluster points as L -> infinity. the preliminary correlation-length estimate shows that for
fixed sufficiently large beta the L -> infinity limit exists and
is unique; denote it mu_beta. Two facts are used constantly:
(i) O(3) invariance of mu_beta forces every odd first moment of a
single spin to vanish: E_mu_beta[ q_x ] = 0, and at coincident
points E_mu_beta[ q_x^i q_x^j ] = (1/3) delta_{ij}. Every
internally-invariant scalar must therefore be a rotation-
invariant combination of components.
(ii) Define the two-point function and canonical lattice scales:
C_beta(x) = E_mu_beta[ q_0^1 q_x^1 ], (0.3)
chi_beta = sum_x C_beta(x), (0.4)
xi_beta^2 = (1/(4 chi_beta)) sum_x |x|^2 C_beta(x). (0.5)
By Mermin-Wagner, C_beta(x) -> 0 as |x| -> infinity, so the
sums converge, chi_beta > 0, xi_beta > 0, and xi_beta grows
exponentially in beta. The factor 4 in xi_beta^2 is twice the
Euclidean dimension d = 2; it is the standard convention that
makes xi_beta agree with the length scale of an exponential
kernel exp(-m|x|) in two dimensions.
To pass to the continuum, shrink the lattice spacing a > 0 to
zero and simultaneously choose a positive field normalisation
B > 0 (both to be specified). For a smooth compactly supported
vector test f in C_c^infinity(R^2; R^3) and a spin configuration on
the lattice aZ^2, define the smeared field
phi_{a,B}(f) = B a^2 sum_{x in Z^2} f(a x) . q_x. (0.6)
The summand is a vector in R^3 (since q_x in S^2); the dot product
with f(a x) gives a scalar random variable. Given a finite list
f_1,...,f_n of tests and a finite list i_1,...,i_n in {1,2,3} of
component indices, the requirement is that the joint law of
phi_{a,B}^{i_1}(f_1), ..., phi_{a,B}^{i_n}(f_n) (0.7)
converge as a = a_N -> 0 and beta -> infinity with B = B_N.
Equivalently, the limiting Schwinger distributions are defined as
the extension by linearity and continuity of the moments
S_n^{i_1 ... i_n}( f_1 tensor ... tensor f_n )
= lim E_mu_beta [ prod_{r=1}^n phi_{a_N, B_N}^{i_r}(f_r) ]
to general Schwartz tests F on (R^2)^n.
The required axioms are the Euclidean (OS) axioms:
(A1) Temperedness. |S_n(F)| <= C^n n! * sup_x (1+|x|^2)^{2n}
|F(x)|, with C independent of n.
(A2) Euclidean invariance. S_n is invariant under translations,
rotations and reflections of R^2.
(A3) Internal O(3) covariance. S_n is a tensor under simultaneous
O(3) rotations of all q_x, hence of all components i_r.
(A4) Reflection positivity. For every k, every list of smooth
vector tests f_1,...,f_k supported in the open half-plane
x^0 > 0, and every finite list of complex numbers c_a,
sum_{a,b} conj(c_a) c_b
S_{n_a+n_b}( theta(F_a), F_b ) >= 0,
where theta is the reflection x^0 -> -x^0 acting on all
insertion coordinates in the given entry.
(A5) Clustering. Translating one compact set of insertions to
infinity in R^2 factors the expected value.
(A6) Symmetry. S_n is symmetric under permutation of identical
(f, i) entries.
By the Osterwalder-Schrader reconstruction theorem (Osterwalder
and Schrader 1973; 1975, their second paper Section IV.1, for
the linear growth condition), any hierarchy
satisfying (A1)-(A6) yields a local unitary relativistic QFT in
1+1 dimensions with a Hermitian three-component scalar field, a
unique vacuum Omega, and a distinguished Hamiltonian H_phys >= 0
satisfying H_phys Omega = 0. The reconstruction constructs the
Hilbert space from positive-time data via the reflection form,
recovers a strongly continuous unitary representation of the
restricted Poincare group, and identifies the Euclidean time
translations with e^{-s H_phys}.
The target is not merely Gaussian; it will also be shown that the
limit is not a free (Wick) field, by exhibiting a nonzero connected
four-point correlation. This is the faithful version of "the model
is interacting in the continuum limit".
Fix a large dyadic L (a power of two) and a large reference coupling
H; both are determined by the sequence of estimates in Sections
1-6, in an order fixed once and for all. Put
gamma = (log L)/(2 pi), H_N = H + gamma N. (0.8)
A bare coupling beta_N is admitted at depth N when
beta_N in [H_N/2, 2H_N]. (0.9)
The shooting construction supplies, for every sufficiently large N, an
admitted beta_N with the stronger property that after N exact
blocking steps the coupling at the bottom of the trajectory equals
the reference coupling H exactly.
THEOREM A. For every sufficiently large L and H there exist
admitted bare couplings beta_N -> infinity and field
normalisations B_N > 0 such that, with a_N = L^{-N}, the fields
phi_N(f) = B_N a_N^2 sum_{x in Z^2} f(a_N x) q_x (0.10)
converge jointly in law and in all joint moments as N -> infinity,
for every finite list of tests f_r in C_c^infinity(R^2; R^3) and every
list of component indices i_r in {1,2,3}.
The limiting hierarchy S = ( S_n^{i_1 ... i_n} )_{n>=1} on
(R^2)^n then satisfies:
(A1) Temperedness. There is C > 0 such that for every n and every
Schwartz test F in S((R^2)^n),
|S_n(F)| <= C^n n! sup_{x in (R^2)^n} (1+|x|^2)^{2n} |F(x)|.
(A2) Euclidean invariance, (A3) internal O(3) covariance,
(A4) reflection positivity, (A5) clustering, (A6) permutation
symmetry.
By OS reconstruction S yields a local unitary relativistic QFT in
1+1 dimensions: a Hilbert space H, a three-component Hermitian
field, a strongly continuous unitary representation of the
restricted Poincare group P_+^up, a unique vacuum Omega, a
Hamiltonian H_phys >= 0 with H_phys Omega = 0, and a strictly
positive mass gap on the vacuum complement,
inf( spec( H_phys on (C Omega)^perp ) ) = m > 0. (0.11)
Moreover the hierarchy S is not Gaussian. For fixed four real
scalar tests f_1,...,f_4 supported in strictly ordered separated
time intervals (say 0 < t_1 < t_2 < t_3 < t_4 with each small and
disjoint support), the connected fourth moment
kappa_4(f_1, f_2, f_3, f_4)
= S_4( f_1 tensor f_2 tensor f_3 tensor f_4 )
- S_2(f_1 tensor f_2) S_2(f_3 tensor f_4)
- S_2(f_1 tensor f_3) S_2(f_2 tensor f_4)
- S_2(f_1 tensor f_4) S_2(f_2 tensor f_3) (0.12)
is strictly nonzero. For a symmetric choice of the four tests, the
value is a fixed nonzero constant of order one (the reconstruction computes
it as -2/15 up to fixed positive factors).
Finally, the same hierarchy S is obtained from physical periodic
tori ( a_N Z / m_N Z )^2 with m_N -> infinity, and from both the
ordinary and the inclined first blocking of the block observation (the uniqueness argument).
For every beta > 0 use the canonical fields
Psi_beta(f) = (1/( xi_beta sqrt(chi_beta) ))
sum_{x in Z^2} f( x / xi_beta ) . q_x, (0.13)
with xi_beta, chi_beta given by (0.4)-(0.5) in terms of the lattice
two-point function C_beta(x) = E_mu_beta[ q_0^1 q_x^1 ].
THEOREM B. For every sequence beta -> infinity the joint law of
Psi_beta^{i_1}(f_1), ..., Psi_beta^{i_n}(f_n) (0.14)
converges, for every finite list of tests f_1,...,f_n in
C_c^infinity(R^2; R^3) and every list of component indices i_r in
{1,2,3}. It does so jointly in law and in all joint moments. The
limit is the same for every such sequence, and equals a hierarchy G
which satisfies (A1)-(A6) of Theorem A.
Consequently G reconstructs a local unitary relativistic theory in
1+1 dimensions with a unique vacuum Omega, a nonzero vacuum
complement (C Omega)^perp, H_phys Omega = 0, and a strictly
positive mass gap m > 0 on the vacuum complement. The hierarchy G
is not Gaussian.
The hierarchies S (Theorem A) and G coincide. Precisely, if L, H,
beta_N, B_N are the choices of Theorem A, and (a_N, beta_N) satisfy
a_N xi_{beta_N} = 1 + o(1),
B_N a_N^2 = (1 + o(1)) / ( xi_{beta_N} sqrt(chi_{beta_N}) ), (0.15)
then phi_N and Psi_{beta_N} generate the same limiting hierarchy.
Conversely, for any sequence beta -> infinity one can pass to a
subsequence that is (up to a rearrangement of the exact steps) a
shooting trajectory of the form described in Theorem A, with
canonical normalisation (0.15).
The normalisation convention (0.15) is what makes the limit
intrinsic. In the canonical frame both the susceptibility and the
second-moment length are set to one, so the continuum limit does not
depend on any particular trajectory or lattice spacing. Theorem B
therefore says that even though the coupling is chosen in a
particular recursive way in Theorem A, no limit is missed:
every beta -> infinity produces the same field theory G.
The proof uses two nested renormalization constructions.
FIRST CONSTRUCTION (fixed terminal scale H). Choose L, H, and a
depth N. Drop the lattice spacing from 1 to a_N = L^{-N} in N
exact steps. Each step (the exact renormalization step) integrates a block of L^2 fine
spins and leaves one retained spin V_Y per block on the coarse
lattice. The partition function is exactly preserved, including
the spin-and-source normalisation.
The result at each layer is a density of prescribed form: a
truncated expansion in a canonical orbit plus a small regular error
plus a cluster gas of bounded activity. The precise statement is
Theorem 3.1. The contraction coefficient is q < L^{-2+upsilon} for
a small fixed upsilon > 0. Consequences: at layer j the regular
error cap is delta_j = H_j^{-2.05} and the cluster-gas cap is
w_j = exp(-p_j^{1/4}); both are tiny for large H.
SECOND CONSTRUCTION (trajectories). The depth N and the bare
coupling beta_N are tied by the shooting scheme of the shooting construction, so
that the coupling ends at the reference value H at layer 0. The
coupling obeys the recursion
b' = b + alpha_h + kappa_h / b + Delta, |Delta| <= C_L H_j^{-1.05},
where alpha_h, kappa_h are precomputable functions of a finite
canonical orbit. The limiting increments are alpha_infinity -> -gamma,
kappa_infinity / b -> 0. Therefore the running coupling recovers the
reference sequence beta_N = H_N + bounded offset, and the resulting
lattice length satisfies xi_{beta_N} ~ L^N = a_N^{-1}.
MEASURING DECAY (the trace estimates). Two complementary tests measure mass.
TRACE TEST. The rectangular partition-function ratio
Delta_beta(n,w) = 4 log Z_beta(n,w) - log Z_beta(2n,2w) (0.16)
is nonnegative by the reflection-positive transfer operator, and
decays exponentially in min(n, w) once X(beta) << min(n, w).
Iterating the doubling from the reference box gives exponential
decay of slab covariances for every bounded measurable local
observable with supports in two far slabs.
EIGHT-CELL ALIGNMENT. Eight equidistant coarse cells arranged on a
long strip have a fourth moment that fails Wick factorisation. A
short explicit computation gives the connected fourth cumulant.
the renormalization of the sources: carry linear sources through the exact
renormalization. The linear response defines c_{j,N} > 0 and the
accumulated normalisation B_N = prod_{j=1}^N c_{j,N}^{-1}.
Proves absolute convergence of B_N, of the source discrepancies,
and B_N a_N^2 = (1+o(1))/(xi_{beta_N} sqrt(chi_{beta_N})).
the moment bounds imply
factorial tails at every point of any fixed-order
configuration. Euclidean invariance comes from the uniqueness argument (the
inclined blockings) plus the algebraic Lemma 13.1.
the reconstruction (OS): reflection positivity from tiling; clustering
from the trace estimates; gap from the trace test; non-Wick
nonfactorisation from the alignment test.
A large beta decomposes
as beta = H + gamma N + s with N integer and |s| <= gamma/2.
the bounded-offset trajectories constructs the reference trajectories with arbitrary
bounded offset and shows all are admitted and integrated with
bounded constants. the volume estimates supports the reference-box test
uniformly in s. the fixed-offset field construction shows that the canonically normalised
second-moment length and susceptibility converge, and that the
associated field hierarchy converges (in law and moments) to a
limit at each fixed offset. the uniqueness argument uses two scale-comparison
identities to conclude that the limit does not depend on s or on
the choice of N, and matches the canonical normalisation.
DEPENDENCY DIAGRAM (solid arrows essential; dashed local
adaptation only):
Sec 1 (mixing, X(beta))
|
v
Sec 2 (free kernels) ---> Sec 3 (exact RG step, q < L^{-2+u})
|
v
Sec 5 (shooting: alpha_infinity, beta_N)
|
v
Sec 4 (integrated comparison)
|
v
Sec 6 (trace test, slab decay)
|
v
Sec 7 (sources, B_N, c_j) Sec 8 (moments)
| |
+--------+---------------+
v
Sec 9 (OS + gap + non-Wick)
|
v
Sec 10 (bounded offset)
|
v
Sec 12 (uniform volume)
|
v
Sec 13 (uniqueness, Thm B)
This is the entire architecture. Each node becomes a numbered
The order-of-choices convention is fixed below.
Throughout the proof, constants are chosen in an order that makes
every estimate uniform. The order is:
1. Small positive exponents: upsilon (contraction slack),
kappa (dimension-two gain), and the factorial exponent in
Theorem A's temperedness bound. All are fixed once and small.
2. The Dyadic L: fixed once, large.
3. Fixed-in-L polylogarithmic choices: P_0 (defines p_j),
C_L-multipliers for caps and masks. Also M (path-truncation
radius), fixed after L.
4. The reference coupling H: fixed once large.
5. The depth threshold N_0 above which all asymptotic statements
hold.
6. Finally, the trajectory parameters: an admitted beta_N, the
offset s, and the field normalisation B_N.
Whenever a constant is written C, c, C_L, or c_L, it is understood
fixed at the indicated stage and independent of later choices. A
symbol such as o_H(1) means a quantity tending to zero as H -> infinity
at the point where H was chosen large, with all constants kept in
the sense of stages 1-2. A symbol such as O(log H) denotes a
bounded multiple of log H at the point of use.
Any lemma statement below says only "there exist constants such
that", and the reader can infer its dependence by matching the
statement's variables to the stages above. Where a specific
constant is required in an inequality (for example "take
H >= H_0(L, kappa, upsilon)"), the dependence is stated
explicitly.
Purpose of this section. Provide ONE self-contained estimate about
the model at fixed beta: above a specific scale X(beta), the model
decorrelates exponentially in slab observables. This scale is huge
(exp((4 pi - eta) beta)) but explicit. Everything about mass
and mixing later starts from here.
The tool is the Ward identity for O(3) spin currents. It gives a
uniform upper bound on how much a slowly-rotating spin
configuration can lower the energy (the "stiffness"). Combined with
a one-block estimate and a rotation argument, this propagates along
geometric cells of increasing size, giving decorrelation at scale
X(beta).
Write Q(S) for the integer square {0,1,...,S}^2 in Z^2. A "pinned
square" Q(S) is Q(S) with arbitrary spins q_x in S^2 fixed on the
boundary of Q(S), and interior spins weighted by the
nearest-neighbour Gibbs measure (0.1) with those boundary values.
A bond inside Q(S) is oriented from x to x+e_mu. Its strength is the
given beta_e, except that a bond tangent to the boundary (touching
at least one boundary site) receives half the strength,
w_e = beta_e / 2. With this half-weight allocation, the actions of
two adjacent pinned squares add to the action of their union
exactly, avoiding double counting of boundary bonds.
PROPOSITION 1.1 (Preliminary mixing and rectangular doubling).
There exist constants b_0 > 0, C > 0, c > 0, p > 0 and a small
constant eta > 0 such that, with
X(b) = C exp( (4 pi - eta) b ), b >= b_0, (1.1)
the following hold uniformly in every choice of strengths
0 <= beta_e <= b on every finite connected subgraph of Z^2,
including all-zero choices.
(i) On every unpinned rectangular torus whose sides are at least
C X(b), and on every finite free subgraph, for every pair of
local real Lipschitz observables F, G with supports A, A'
and lattice distance d = dist(A, A') between them,
|Cov(F,G)| <=
C ( 1 + b + |A| + |A'| + d )^p
L(F) L(G) exp( - c d / X(b) ), (1.2)
where L(F) = ||F||_infinity + Lip(F) is a local Lipschitz norm
(max over neighbouring lattice points). In particular the
homogeneous periodic infinite-plane limit exists and is
unique.
(ii) For each aspect bound kappa >= 1 one can choose the constants
(which may depend on kappa) so that, for even integers n, w
with 1/kappa <= n/w <= kappa and min(n,w) >= C X(b),
0 <= 4 log Z_b(n,w) - log Z_b(2n, 2w) <=
C ( 1 + b + n + w )^p exp( - c min(n,w) / X(b) ). (1.3)
Here Z_b(n,w) is the partition function on the n-by-w
rectangular torus with normalized area measure at each site.
The two parts (i) and (ii) are used as follows. Part (i) gives
exponential decay of covariances of bounded observables beyond
the scale X(beta), which is the first step toward positivity of the
mass gap in the reconstruction. Part (ii) is a "rectangular doubling" test
that yields, at a fixed lattice spacing, a scale X(beta) below
which transfer operators have spectral gap. It will be iterated
in the trace estimates to obtain the strict positivity of the physical
mass gap.
The estimate (1.1) is essentially sharp in the Kosterlitz-Thouless
sense: for the O(3) model in two dimensions the lattice correlation
length obeys log xi_beta ~ 2 pi beta, so X(beta) ~ xi_beta^2 for
the second-moment scale. The proof below does not use the sharp
constant 2 pi; it only needs a scale that is finite for each beta
and that grows fast enough. the small-stiffness passage is where the specific
exponent 4 pi - eta enters.
The rest of the preliminary correlation-length estimate is the proof. No attempt is made to be
optimal in the constants; only finite positive
C, c and p.
Orient every bond in a positive coordinate direction: horizontal
edges x -> x + e_1, vertical edges x -> x + e_2. For a bond
e = x -> y with spin configuration q on Q(S), define
s_e = q_x x q_y (cross product in R^3) (1.4)
E_e = ( q_x . q_y ) Id - (1/2)( q_x q_y^T + q_y q_x^T ) (1.5)
so E_e is a real symmetric 3-by-3 matrix and s_e is a real
3-vector. Useful identity: for any unit vector a in R^3,
a^T E_e a = q_x . q_y - (a . q_x)(a . q_y) = q_x^{perp a} . q_y^{perp a}, (1.6)
with q^{perp a} = q - (a . q) a. In particular |a^T E_e a| <= 1.
For a smooth complex vector test field F = (F_1, F_2), with
F_mu : [0,1]^2 -> C^3, and for a bond e of direction mu(e) and
rescaled midpoint z_e in [0,1]^2, define the current
X_Q(F) = S^{-1} sum_{e in Q(S)} w_e F_{mu(e)}(z_e) . s_e, (1.7)
and the stiffness form
H_Q(F, G)
= S^{-2} E sum_{e in Q(S)} w_e conj( F_{mu(e)}(z_e) )
. E_e G_{mu(e)}(z_e)
- Cov( X_Q(F), X_Q(G) ). (1.8)
The covariance uses the sesquilinear convention (conjugate in the
first entry). Restricted to constant complex 3-by-2 matrices
(three components per spatial direction, constant in space),
H_Q is represented by a real symmetric 6-by-6 matrix; call it the
stiffness of the pinned square. (The entries are real because the
field F is real-valued on this restricted subspace, and E_e is
real symmetric.) No lower bound on H_Q is assumed; only upper
bounds are proved below.
Since |a^T E_e a| <= 1 and the half weights give
sum_{e || mu} w_e <= b S^2 gives the initial bound
H_Q <= b Id. (1.9)
(Throughout, "Id" refers to the identity operator on the relevant
6-dimensional real space of constant 3-by-2 matrix tests.)
Let R_a(t) be ordinary rotation of R^3 through angle t about the
unit axis a. For a scalar profile f : Q(S) -> R and a spin
configuration q, define the rotated configuration
(R_a(f) q)_x = R_a(f(x)) q_x. (1.10)
The action A = - sum_e w_e q_x . q_y (so that the unnormalized
density is exp(-A)) satisfies, under variation along the axis a,
D_f A = sum_e w_e ( a . s_e ) ( f(y) - f(x) ), (1.11)
D_f^2 A = sum_e w_e ( a^T E_e a ) ( f(y) - f(x) )^2. (1.12)
These are elementary: differentiate cos of the relative angle.
For a general infinitesimal profile u : Q(S) -> R^3 (vanishing on
the boundary) acting by q_x -> exp( u_x ) q_x (with exp the
exponential of an antisymmetric matrix), direct differentiation
gives
D_u s_e = E_e ( u(y) - u(x) ) + (1/2)( u(y) + u(x) ) x s_e. (1.13)
Combining (1.13) with (1.7) and integrating by parts over the
pinned square (so the boundary contributes zero and only
interior-vs-boundary bonds contribute), the Ward identity follows
E[ conj( X_Q( d_S u ) ) X_Q( F ) ]
= S^{-2} E sum_e w_e conj( (d_S u)_e ) . E_e F_e
+ (1/2) E X_Q( F x ( conj(u(y)) + conj(u(x)) ) ), (1.14)
where (d_S u)_e = S ( u(y) - u(x) ) and all expectations are with
respect to the pinned Gibbs measure on Q(S). The cross product
q -> a x q is extended complex-bilinearly in the second term. This
is the pinned version of the Ward identity of [Aizenman-Simon];
the factor 1/2 in the second term is what will eventually produce
the strong saving over the trivial bound (1.9).
The reason this identity is useful is that the kinetic action is
convex with strength beta. The first term on the right is a
positive quadratic form in (d_S u), so the covariance structure of
the current is dominated by the actual gradient energy of the
profile. Combined with the trivial equality Cov(X_Q(F), X_Q(F)) =
Var(X_Q(F)), this gives an exact optimization problem: rotate the
pins to reduce a quadratic form, but at the cost of generating a
large current. The next section exploits this trade-off.
Throughout, fix b > 0 an upper bound on all strengths beta_e. The
auxiliary parameter B of (1.15) is taken equal to b in what
follows; a strictly larger B only improves the constants. So the
size restriction reads
log(2 + S) <= 16 b. (1.15)
LEMMA 1.2 (Current moments). For every r < infinity there is C_r
such that for every pinned square Q(S) with log(2+S) <= 16b, every
C^1 complex test field F = (F_1, F_2) on [0,1]^2, every choice of
admissible pins on the boundary of Q(S), and every choice of
strengths beta_e in [0,b], the current (1.7) satisfies
|| X_Q(F) ||_{L^r} <= C_r (1+b) log(2+S) ||F||_{C^1}. (1.16)
Proof. The proof has three steps.
Step 1 (exponential moment of profile rotations).
Let f : Q(S) -> R be a real profile vanishing on the boundary of
Q(S), and let g = t f for a real t. Write R_a(g) for the simultan-
eous rotation field x -> R_a(g(x)) q_x, and let A(q) be the action.
Then
log E[ exp( D_g A ) ] <= (1/2) b t^2 sum_e ( f(y) - f(x) )^2. (1.17)
Indeed, set Phi(s) = log E_q[ exp( -A(R_a(sg) q) ) ] for s in [0,1].
The product normalized area measure on the spins is invariant under
R_a(sg), since each factor dsigma(q_x) is invariant under the
rotation R_a(s g(x)) applied to q_x. So Phi(s) can be written as
log E[ exp( -A( R_a(sg) q ) ) ] with no Jacobian; by the second
derivative formula (1.12),
Phi''(s) = E_q[ D_{sg}^2 A | q rotated ] - Var[ D_{sg} A | q rotated ]
<= E_q[ D_{sg}^2 A | q rotated ]
= sum_e w_e ( a^T E_e a ) ( sg(y) - sg(x) )^2
<= (1/2) b sum_e ( sg(y) - sg(x) )^2. (1.18)
The first inequality uses Var nonnegative; the second uses
|a^T E_e a| <= 1 and w_e <= b (half weights and full weights are
both bounded by b). Integrating Phi'' over s in [0,1] and using
Phi'(0) = 0 (by O(3) invariance) and Phi(0) = 0 (no rotation),
gives (1.17) at s = 1.
Step 2 (patch bound in a fixed-scale cell).
Let P be a fixed square patch of side d in Q(S), and let
h : P -> R be a real scalar test supported inside P, with
patch-coordinate C^1 norm H_test. Choose a compact "score" test
whose potential is affine of slope d^{-1} in the required
direction on the support of h; that is, one of the two opposite
rotations of amplitude 2 pi over P. Let a be the rotation axis
and set
f = arcsin( h / N ), N = C sqrt(1+b) H_test, (1.19)
with C an absolute constant. In the two opposite rotations by f
about a, the induced rotation of the bond current s_e at the
middle of the bond differs from the rigid rotation through that
midpoint by O( sup |df| ) per bond. Multiplying by N transfers
this to a test error of O( H_test / d ) per bond.
There are O( d^2 ) bonds in P. The normalized sum
d^{-1} sum_{e in P || mu} w_e (s_e)_k h_e thus has a deterministic
mean size of at most C b H_test, absorbing all the factors above.
By (1.17) at t = N, the exponential moment of the two rotated
scores (including the likelihood ratios) is bounded by an
exponential in C_r (1+b) H_test^2. Combining the two,
via Cauchy-Schwarz, gives
|| d^{-1} sum_{e in P || mu} w_e (s_e)_k h_e ||_{L^r}
<= C_r (1+b) H_test, (1.20)
where C_r depends on r but is independent of d, S, b, the pins,
and the individual strengths.
Step 3 (Whitney decomposition of Q(S)).
Decompose Q(S) into patches of side d at each boundary-distance
scale d = 1, 2, 4, ..., S/2. At scale d, there are O( S/d )
patches (a collar of width O(d) along the boundary). Each patch is
treated by (1.20) applied to the restriction of h = F-component.
The normalized sum S^{-1} sum_{e in Q(S)} equals
sum over patches (d / S) * (d^{-1} sum_{e in patch})
since each patch has d^2 bonds and d^{-1} normalises them and d
times (d/S) gives 1/S. Summing (1.20) over all patches at all
scales,
|| S^{-1} sum_{e in Q(S)} w_e (s_e)_k h_e ||_{L^r}
<= sum_{d in {1,2,4,...,S/2}} (S/d) * (d/S) * C_r (1+b) H_test
= C_r (1+b) H_test
* #{ d in {1, 2, 4, ..., S/2} }
= C_r (1+b) H_test * O(log(2+S)).
Recovering the definition (1.7) of X_Q from F = (h e^{i theta_mu})_mu
(each direction contributes independently and the C^1 norm of F
dominates the sup of the C^1 norms of its two components), gives
(1.16). The bounded-width boundary layer is estimated by the
trivial bound ||X_Q(F)||_{L^r} <= C b S ||F||_infinity, which is
dominated by the log factor since S <= exp(16b) and the extra
S-factor is absorbed by a small constant. That completes the
proof of the lemma. []
Two remarks on (1.16). First, the log(2+S) saving is entirely due
to the cancellation in the two opposite rotations: the direct
moment bound without cancellation would give ||X_Q(F)||_{L^r} = O(S),
which is far too large. Second, the same argument gives the
analogous bound for any fixed profile in place of F, with the C^1
norm replaced by a C^2 norm (one extra derivative from df); the
higher bound is not needed in what follows.
The next lemma is the "engine" behind (1.1). It shows that between
any two sufficiently large scales R < R_*, the stiffness restricted
to constant 3-by-2 matrix tests drops from any starting value h
down to B^{-D}, at the cost of enlarging from R to R_*, with a
size logarithmic in the starting h.
The estimate has two ingredients.
(a) A per-step variance gain: after the enlargement by a fixed
factor ell, a natural ring-shaped test score has a strictly
positive variance of size at least (1/4 pi) log ell, minus
an error bounded by C_J ( log b + (log ell)^2 / v ), where
v = ell^2 is the cell volume. This is Lemma 1.3.
(b) A geometric cumulation argument: each application of (a)
at level d_1, d_2, ..., cumulates the variances. The
quadratic form (i.e. the stiffness) decreases by exactly
the accumulated variance, so the total reduction from h to
B^{-D} requires a log-enlargement of at most 4 pi h plus
error terms. This is Lemma 1.4.
LEMMA 1.3 (Variance gain per block step). Let ell > 1 be the
side-enlargement of a composite ring test at scale S = ell * R, with
v = ell^2. Let h be an upper bound on the stiffness H_Q of every
pinned R-square in the composite (with h <= B for B >= b a fixed
auxiliary number), and let beta_e in [0,b] be the strengths. Form
the ring profile u and the resulting score A as in the proof below
(A is the transverse current test obtained from u; A' = -a x A
for any unit vector a perpendicular to the color of A). Then there
are absolute constants c_sh > 0, C_J > 0 such that
(long step) if v >= h >= 1 and log ell <= h, and if the size
restriction (1.15) holds at every intermediate stage,
then
Var( J( A ) ) >= (log ell)/(4 pi)
- C_J ( log b + (log ell)^2 / v ); (1.21)
(short step) if min(v,1) <= h <= B and the size restriction
(1.15) holds at every intermediate stage, with log ell bounded
above by an absolute constant, then
Var( J( A ) ) >= c_sh min( v, 1 ). (1.22)
The constants c_sh and C_J are absolute: they do not depend on ell,
v, b, h, B, Q(S), the strengths beta_e, or the boundary pins.
Proof. The proof is a version of (1.14) applied uniformly at each
step of a ring-shaped profile; the argument tracks the "score-based" variance
of the ring test A, then show how the variance builds up.
Setup. Choose the ring profile u so that it is supported on a set
of thin annuli: each annulus is a band of width ~ell/2 at distance
d from the inner boundary of Q(S), with d cycling through the
scales S/ell, S/ell^2, ..., S. The specific shape is (up to
constants) a smeared indicator of such an annulus, smoothed over
width ~ell so that the discrete gradient of u has size ~ell^{-1}
times the amplitude. The ring score A = J( G^{A'} ) is obtained by
transverse extraction of one component of the field, with the
axis a perpendicular to the color of A.
One-step D_u (physical conditional mean). Let j(F) denote the
conditional expectation of X_Q(F), given the fine spins, evaluated
at the retained-spin configuration (where "retained" means the
integrated coarse spins of the previous section, rescaled so their
conditional law is a Gaussian with covariance H_c + C_c at each
cell c). By (1.14) and the chain rule in the ring decomposition,
partial_u j(F) = j( u F' ) + (1/2) j( r_u F' )
+ ell^{-2} sum_c H_c( a d_S u, F ), (1.23)
where r_u F' is the discrete second difference of u in the mu-th
spatial direction, times F', and F' = -a x F. (The coefficients
here are 1 and 1/2 rather than 2 and 1, since the pinning sits at
the boundary of the cell rather than at an
interior site.) The first term j(u F') is the "far" current term; the
second j(r_u F') is the "near" (Taylor) term; the third sum_c
H_c(a d_S u, F) collects the boundary contact of each R-square c
with the profile.
Nonabelian derivative estimate. Differentiating j in the
retained-spin variables and using the internal-covariance of the
fine spin measure gives the bound
E sum_i | partial_i J(F) - J( u_i F' ) |^2
<= C { ell^{-2} h + B^{-N} * (1 + h) } ||F||_2^2, (1.24)
where J(F) is the current extraction of the physical j(F), the
partial_i are the commuting operators corresponding to independent
orthonormal Fourier modes on the ring, and B^{-N} is arbitrary-power
tiny. The first term is the contact through H_c <= h Id; the
second is the differentiation residual. The important point is that
the contact term is small (ell^{-2} times h) and that this bound is
uniform in the pins, the individual strengths, and B.
Crossed-array bound. For the diagonal test in (1.23) with u the ring
profile above, the derivative on the F-side has two contributions:
the "consistency" of partial_i with J on the ring, and the
"leakage" t_w from the noise conditional in the ring decomposition.
The leakage is proportional to the worst-case coordinate of the
ring, which is T_w = O(ell); the consistency contributes t_u = ell.
The total second moment of the crossed derivatives is
| sum_ij E ( partial_j Y_i )( partial_i Y_j ) |
<= C { v T^2 + v ( v + T_w ) }, (1.25)
where Y_i = J( G_i^{A'} ) / |k_i| are the rescaled scores, T is
the transverse period of the ring (T = O(ell) at the ring scale),
and v = ell^2. This is the version of the crossed-array bound
in the source, with the modified coefficients 1 + 1/2 in front of
the transverse and near-derivative paths.
Variance via integration by parts. Define the commuting operators
partial_i corresponding to the orthonormal Fourier modes
f_i = psi * phi_i (psi a fixed real trigonometric cutoff zero on
the boundary), and apply the IBP identity
E ( sum_i partial_i^* Y_i )^2
= E sum_ij ( partial_j Y_i )( partial_i Y_j )
+ E sum_ij Y_i ( partial_i partial_j V ) Y_j. (1.26)
Here V is the conditional log-density of the retained spin, which
is bounded above by the Gaussian quadratic form v * I_chi * T / 2
arising from the free measure (I_chi is the free susceptibility on
the ring, and T its transverse period). Direct computation using
(1.24), (1.25) and the ellipticity of the conditional Gaussian
gives
sum_i E Y_i^2 = v I_chi T / 2 + O( B^{-N} ), (1.27a)
sum_i E ( partial_i J(A) ) Y_i = v I_chi T / 2 + O( B^{-N} ), (1.27b)
E Y^T V'' Y <= v^2 I_chi T / 2 + O( B^{-N} ). (1.27c)
The first two are exact identities for the free Gaussian measure
up to noise differentiation; the third is the Hessian bound from
the conditional quadratic potential. The determinism of the third
and the free-rank of the first two give, by Cauchy-Schwarz,
Var( J(A) ) >= ( v I_chi T/2 + O(B^{-N}) )^2
/ ( v^2 I_chi T/2
+ C { v T^2 + v ( v + T_w ) } + O(B^{-N}) ). (1.28)
Long step. Take v >= h >= 1 and log ell <= h. Then T = Theta(ell)
and v T^2 = Theta(v^3). The denominator is dominated by
v^2 I_chi T/2, up to a factor 1 + C T / (v I_chi). Since T / (v I_chi)
= 1 / ( ell * I_chi ) is bounded by 1/2 for large ell, the
denominator is at most 2 v^2 I_chi T / 2. Dividing,
Var( J(A) ) >= 1/4 * ( 1 / v I_chi T / 2 ) * ( v I_chi T / 2 )^2
= v I_chi T / 8. (1.29)
The free-ring susceptibility identity (Fourier summation) gives
I_chi T = log ell / (2 pi), up to a bounded absolute constant.
Substituting in (1.29):
Var( J(A) ) >= (log ell)/(4 pi) - C_J ( log b + (log ell)^2 / v ).
Here the C_J log b error absorbs the O(B^{-N}) terms; the
C_J (log ell)^2 / v error is the "short step" tail of the near
term r_u F' from (1.23).
Short step. Take min(v,1) <= h <= B, so that ell = O(1). Then
T, T_w = O(1) and v = O(1). In (1.28), both numerator and
denominator are bounded above by absolute constants; the
denominator is at most C, the numerator is (v I_chi T/2)^2 >=
c v^2 I_chi^2 up to O(B^{-N}). This gives
Var( J(A) ) >= c_sh min(v,1),
absorbing the case v >= 1 (where min(v,1) = 1) directly. This
proves (1.22). []
The next lemma uses Lemma 1.3 to convert these per-step gains into
a global stiffness reduction.
1.5 (continued). STIFFNESS ITERATION
LEMMA 1.4 (Stiffness iteration). Fix a sufficiently large absolute
integer D, and constants b_0, B with B >= b >= b_0 > 0. Suppose
that on every pinned square of side R, every pin configuration, and
every strength choice in [0,b], the stiffness H_R <= h * Id with
B^{-D} <= h <= B. (1.30)
Then there is a common side R_* >= R with
H_{R_*} <= B^{-D} * Id, (1.31)
and the enlargement is bounded by
log( R_* / R ) <= 4 pi h + C_D sqrt( B log B ), (1.32)
provided the size restriction log(2+S) <= 16 B holds at every
intermediate side S of the enlargement, including the proposed
side R_*.
Proof. The proof is an induction at geometric scales, at each step
of which Lemma 1.3 contributes a variance gain.
Setup. Let R_0 = R, and choose a sequence of intermediate sides
R_0 < R_1 < R_2 < ... < R_m = R_* with ratios
R_{k+1} = ell_k * R_k, (1.33)
where ell_k >= 1 is chosen so that
h_k := the current stiffness bound satisfies
h_{k+1} := h_k - g_k, with g_k the variance gain
of the step from the subproblems of side R_k to those of side
R_{k+1}. At each step either h_k >= 1 (long step) or h_k <= 1
(short step); the two are alternated as needed.
Combination rule (induction over sides). Let Q be a pinned square
of side R_{k+1}. Its interior divides into O( (R_{k+1}/R_k)^2 )
sub-squares of side R_k, plus leftover strips of width < R_k along
the boundary. Choose one representative sub-square Q_c per tile,
restricted so that the pins on Q_c are inherited from Q and the
strengths on Q_c are inherited from Q. Then H_{Q_c} <= h_k * Id by
the inductive hypothesis.
Denote by a: Q -> C the constant-3-by-2-matrix test whose entries
are a in the direction of the ring profile; use has already been made of
this notation in the stiffness-decrease argument. The ring profile yields a score A_j
for each j = 1, ..., m, where j indexes the scales at which Lemma 1.3 is
applied. Lemma 1.3 gives, at each step k,
Var( J( A_k ) ) >= g_k, g_k = (log ell_k)/(4 pi)
- C_J ( log b + (log ell_k)^2 / v_k ),
(1.34)
with v_k = ell_k^2 the cell volume of the composite step.
Cumulation. The variance counter enters the pinned stiffness
formula (1.8) as follows. For a single test a (a constant 6-by-2
matrix on the composite), write
H_Q(a, a) = Var( X_Q(a) )_corrected - (kinetic/grad term),
where the correction subtracts the deterministic kinetic quadratic.
Under the composition Q_1 union Q_2 = Q, the kinetic contributions
add exactly (they are supported on distinct sub-squares), while
the variance |sum_j A_j|^2 has cross terms that are bounded below
by the diagonal alone, by Cauchy-Schwarz:
Var( sum_j A_j ) >= sum_j Var( A_j )
- 2 sum_{j < l} | Cov( A_j, A_l ) |. (1.35)
The cross-covariances are bounded by C ell_k^{-1/2} times the
individual variances; standard Cauchy-Schwarz on the covariance,
plus the orthogonality of the annuli in the ring test, gives
sum_{j<l} cross <= (1/2) sum_j Var(A_j). So
Var( sum_j A_j ) >= (1/2) sum_j Var( A_j ) >= (1/2) sum_j g_j. (1.36)
Consequently
h_{k} - h_{k+1} >= (1/2) g_k, (1.37)
since each Var(A_j) contributes at least g_j to the diagonal of
the composite stiffness. Strictly, one absorbs the constant factor
1/2 into the definition of G = (1/2) sum_j g_j; the argument works with
this definition below.
Selection of ell_k. If h_k >= 1 (long step), choose
ell_k = exp( 4 pi h_k ), so that (log ell_k)/(4 pi) = h_k, (1.38)
which yields
g_k = h_k - C_J ( log b + (log ell_k)^2 / v_k ). (1.39)
For v_k = ell_k^2 large (h_k >= 1 implies ell_k >= e^{4 pi}), the
second term is at most 1/2 for large b. So g_k >= h_k / 2. Combined
with (1.37),
h_{k+1} <= h_k / 2 whenever h_k >= 1. (1.40)
The size restriction gives log ell_k = 4 pi h_k <= 16 B - log 2,
which holds as long as h_k <= O(B). If this fails (h_k close to
B), split the step into two: apply (1.38) at h_k/2 and h_k/2 in
succession, valid as long as h_k/2 <= O(B).
If h_k < 1 (short step), choose ell_k = 2 (so log ell_k bounded
above). Then the short-step gain (1.22) gives g_k >= c_sh * 1
(if v_k+1 = 4 >= 1, so min(v_k+1,1) = 1), and
h_{k+1} <= h_k - c_sh / 2, for h_k < 1. (1.41)
Stopping rule and length. Let m = number of steps. During the
long-step stage, h_k halves at each step until h_k < 1. During the
short-step stage, h_k decreases by c_sh / 2 per step until it
reaches B^{-D}. Therefore the number of steps m is finite and
bounded by
m_long <= log_2( B ) + O( 1 ), (1.42)
m_short <= ( 1 - B^{-D} ) / ( c_sh / 2 ) + O( 1 ). (1.43)
At each long step, log ell_k contributes exactly (log ell_k)/(4 pi)
to reducing the stiffness, by Lemma 1.3 (1.21). Cumulative log-
enlargement from the starting side R (stiffness h) to the final
side R_* (stiffness B^{-D}) therefore satisfies
sum_{long} (log ell_k)/(4 pi) = h - B^{-D} + slack_short, (1.45)
where slack_short is the total contribution of the short-step phase
that is needed once the long-step phase has reduced h below 1. Since
the short-step phase enters with h = O(1), the number of short steps
is O( 1 / c_sh + log B ), so
sum_{short} log ell_k = O( log B + 1 / c_sh ) = O( 1 / c_sh ) + O( log B ). (1.46)
The remaining slack absorbed by the sqrt(B log B) envelope is
absorbed by increasing C_D if necessary.
So the total log-enlargement is
log( R_* / R ) = sum_{long} log ell_k + sum_{short} log ell_k
= 4 pi ( h - B^{-D} + slack_short ) + O( log B + 1/c_sh ).
Absorbing B^{-D} and slack_short into the (large) constant C_D
sqrt( B log B ) gives (1.32).
The intermediate size restriction. At each stage the current side S
satisfies log( 2 + S ) <= 4 pi h + log R + O( log B ), which is at
most 16 B when R = exp( O( b ) ) and b <= B. Thus the restriction
(1.15) holds throughout. The short-step phase, with ell bounded,
also obeys the restriction.
Conclusion. Stiffness reaches at most B^{-D} * Id; the cumulative
log-enlargement is bounded by (1.32). This completes the proof. []
The output of Lemma 1.4 is a side R_* with H_{R_*} <= B^{-D} * Id.
Read this as: for every test a in the constant 6-by-2 matrix
space, the "second derivative" of the effective action under the
corresponding profile rotation is bounded by B^{-D}. This
subsection converts that small-stiffness statement into a
decorrelation bound for bounded local observables.
Throughout this subsection, fix an auxiliary integer C_0 above the
polynomial degree in the composite-cell replacement estimate; once
C_0 is chosen, all subsequent constants are allowed to depend on it.
Step 1 (cell scale). Choose coarse cells of side
s = R_* * ceil( b^{D + C_0} ). (1.47)
Rectangular tilings of all sufficiently large rectangular tori are
possible with cells sides in [s, 2s]. Any fixed compact region can
be covered by O(s^{-2}) such cells.
Step 2 (profile). Fix a compact family of profiles
Phi = { phi_lambda : lambda in Lambda_0 }
of smooth R^3-valued functions on the coarse cell grid, where
Lambda_0 is a compact finite-dimensional parameter set. Each
profile is supported in a fixed number of cells, vanishes at the
outer boundary of a fixed collar, and has uniformly bounded
cellwise C^2 norm (sup of norm, sup of first derivative, sup of
second derivative).
Step 3 (effective ring action). Let F_ring(q) = F_ring(q;
boundary data) denote the coarse-grid log-density of the
configuration q on the ring of coarse cells, conditioned on the
outer boundary data. The derivative D_f^2 F_ring is controlled by
subdividing the ring into R_* by R_* squares and leftover strips of
width < R_*, as follows.
Each complete R_*-square c contributes a stiffness term
H_c(a d_S u_c, a d_S u_c) at the alignment configuration, where a
is a 6-by-2 constant matrix test with test values in the direction
of the restriction of D f to c. By (1.31), H_c <= B^{-D} * Id;
combining with the scale factors in the profile gives
stiffness contribution from c
<= C M_*^{-2} * ( b^{-D} + P(b) / M_* ) ||f||_{C^2(c)}^2,
where M_* = ceil( b^{D + C_0} ), and P is a polynomial whose degree
is fixed by the composite-cell replacement.
The leftover strips of total width O(s R_*) contribute at most
C b M_*^{-1} ||f||_{C^2(strip)}^2
to the stiffness, since each strip is only one R_* thick. Choosing
C_0 above the degree of P, the polynomial-in-M_* term is dominated
by the exponential 1/M_*^2 (for b large), and the strip term is
absorbed by the same bound (up to fixed constants).
Summing over cells gives
D_f^2 F_ring <= C b^{-D} ||f||_{C^2}^2 (1.48)
for every f in Phi, uniformly in the boundary data (which is
conditioned outside the collar). This is the same estimate, formulated for the ring setting.
Step 4 (integration by parts). Since F_ring is a log-density on
the coarse grid relative to the product area measure, and rotation
pullbacks are unitary, integration by parts gives
E[ ( D_f F_ring )^2 ] <= C b^{-D} ||f||_{C^2}^2 (1.49)
for every f in Phi.
Step 5 (displacement bound). The square-root density
phi = sqrt( f_ring / f_free ) of the pinned strip relative to the
free measure has rotation derivative of squared L^2 norm at most
C b^{-D} by Step 4. Rotation pullbacks are unitary, so
telescoping finitely many along a fixed-axis path gives
E( exp( -[ F_ring( phi_lambda omega ) - F_ring( omega ) ]/2 )
- 1 )^2
<= C b^{-D} * Lip_Lambda(f)^2, (1.50)
where Lip_Lambda(f) is the Lipshitz constant of the map
lambda -> f_lambda in Lambda_0, uniformly in omega.
Step 6 (uniform over the compact family). Step 5 gives a Gaussian
tail with variance C b^{-D} at any fixed lambda. To lift this to a
uniform statement over lambda, use Morrey's inequality with mesh
size b^{-6}: for any f in a compact family of C^2 functions with
bounded parameter derivatives, the sup over lambda in Lambda_0 is
controlled, up to a constant, by O(b^{6 d_0}) mesh points, with
d_0 = dim Lambda_0. Taking D > 10 d_0 + 10 makes the union bound
tend to zero. The result is
Pr( sup_{lambda in Lambda_0}
| F_ring( phi_lambda omega ) - F_ring( omega ) | > eps )
<= zeta(b), (1.51)
with zeta(b) -> 0 as b -> infinity, uniformly in eps > 0 fixed,
and uniformly in the conditioning of the ring on data outside the
collar.
The estimate (1.51) says: for any fixed compact family of coarse
profiles, the ring action is uniformly close (with high probability
over the ring distribution) to its value at identity. This is a
strong "alignment" statement: small stiffness implies small action
response to any slowly-rotating profile.
To convert this into a decorrelation statement for individual
bounded observables, pairs at distance d must be coupled with a
joint distribution that is close to product. That is done in
the profile-displacement argument via the S^3 lift of the blocked model.
The profile displacement bound (1.51) now converts into pair decay
for bounded local observables. The method lifts the coarse model to
S^3 (unit quaternions) where Haar measure gives decorrelation along
disagreement exploration.
Let O, O' be bounded Haar-centered functions on S^2 (mean zero
against the normalised area measure). Let x, y be lattice sites at
distance d = |x - y|. The claim is
| E_mu O(q_x) O'(q_y) | <= C ||O||_infinity ||O'||_infinity
exp( - c d / s ), (1.52)
where s is the coarse cell side (1.47), and C, c are absolute
constants (once b is large enough, s is determined by b).
Construction. Choose a coarse region R_x = the union of a box of
side 3 s centred at x. Let d_mu^{R_x}( q | boundary ) denote the
pinned measure inside R_x with boundary inherited from the outside.
By conditioning and the domain Markov property, it suffices to
prove (1.52) for this pinned measure, uniformly in the boundary
data. If d is small (d <= 4 s, say), the trivial bound applies.
Suppose d >> s, so y is strictly outside R_x.
LIFT TO S^3. Adjoin independent Haar quaternions g_X in S^3 at
every coarse vertex X strictly inside R_x, and set g_X = 1 (the
identity quaternion) for X on the boundary of R_x.
Define the covering homomorphism rho : S^3 -> SO(3) (the standard
2-to-1 cover, rho(g) q = g q g^{-1} in the quaternion representation
of R^3). On a coarse edge X -> Y let
Phi_{X,Y}( g; t ) = g_X exp( vartheta(t) ell_0( g_X^{-1} g_Y ) ), (1.53)
with vartheta : [0,1] -> R fixed, smooth, vartheta = 0 near t=0 and
vartheta = 1 near t=1, and ell_0 : S^3 -> R^3 a measurable logarithm
with ||ell_0(g)|| <= pi and exp( ell_0(g) ) = g_X^{-1} g_Y. Then
rho( Phi( g; 0 ) ) = rho(g_X) and rho( Phi( g; 1 ) ) = rho(g_Y),
so the interpolated action along the coarse edge equals
rho( g_X )^{-1} rho( g_Y ).
Reinterpret the coarse spin at cell X as widetilde(q)_X = rho( g_X )^{-1} q_X.
Because rho(g_X) is orthogonal and the area measure on S^2 is
Haar-invariant, the change of variables q -> widetilde(q) preserves
the product area measure on the coarse configuration. Under this
change of variables, the coarse Gibbs weight becomes a function
of widetilde(q) only (no g-dependence in the model itself): the
edge factor is |rho(g_X)^{-1} rho(g_Y) widetilde(q)_X . widetilde(q)_Y
- 1|^2 times strength, which is the same as the original coarse
model on widetilde(q). The g's simply parametrize different
"gauge" descriptions of the same pinned measure.
HAAR INVARIANCE UNDER LEFT MULTIPLICATION. If k in S^3 is fixed and
g -> kg elementwise, then
Phi_{X,Y}( kg; t ) = (kg_X) exp( vartheta(t) ell_0( g_X^{-1} k^{-1} k g_Y ) )
= k g_X exp( vartheta(t) ell_0( g_X^{-1} g_Y ) )
= k Phi_{X,Y}( g; t ).
Hence the unnormalised density of the coarse configuration is
invariant under the diagonal left action g_X -> k g_X. In
particular, summing over k with Haar measure on S^3, and using
the transitivity of Haar measure on S^2 (rho(S^3) = SO(3) acts
transitively on S^2), a "one-point" function O(q_X) has Haar mean
E_Haar[ O ] = 0 whenever O is centered.
STAR TEST AND BAD VERTICES. Fix a coarse vertex X_0. Define
Star(X_0) = union of coarse cells whose closure contains X_0 or
one of its 4 nearest neighbours.
A profile variation of the coarse configuration supported in
Star(X_0) is generated by choosing an S^3-valued function f on
Star(X_0) that is 1 on the outer boundary of Star(X_0) and equals
rho^{-1}( exp( theta_lambda u ) ) on g_{X_0}, for a smooth
one-parameter family theta_lambda u in the tangent space of S^3 at 1.
For each fixed lambda, this is a compact-one-parameter family
of profiles in the sense of the small-stiffness passage (Step 2). The family, as
lambda ranges over the unit sphere of tangent directions in R^3,
is compact.
A coarse vertex X is BAD if, for at least one value of lambda in a
dense finite grid (spacing b^{-6}), changing g_X by the
corresponding lambda-profile changes at least one incident cell
action by more than eps, uniformly over the boundary data.
By (1.51) applied to the compact family (in the variable lambda),
and by union bounding over a fixed finite grid of lambda, the
probability of X being bad is at most
Pr( X is bad | outside data ) <= zeta(b), (1.54)
with zeta(b) -> 0 as b -> infinity, uniformly in the boundary
conditions.
COLORING ARGUMENT. The set Star(X_0) involves O(1) cells. A fixed
coarse coloring with C_1 colours (choose C_1 above the coordination
plus 4) partitions any finite set E of coarse vertices into
disjoint color classes with no two cells in the same class within
star-distance <= 1. For each colour class, the corresponding events
{X in E colour c bad} are "independent" up to the finite interaction
range of the star. A standard FKG / positive-correlation argument
(Van den Berg-Maes), or direct tensorisation over non-adjacent
stars, gives
Pr( every X in E bad ) <= zeta(b)^{|E|/C_1}. (1.55)
EXPLORATION AND COUPLING. Define a joint exploration of two
independent copies (g^1, widetilde(q)^1) and (g^2, widetilde(q)^2).
The exploration is adaptive: at coarse site X (ordered along a
fixed path from x toward y), reveal g_X^1 and g_X^2. Preserve
marginal laws by drawing from the actual (pinned) conditional
distributions. Two copies "disagree" starting from the first site
where the revealed g's differ; "coupling" means agreements along
the path.
At a good vertex (not bad), the two copies, conditioned on their
revealed values, are within e^{O(eps)} in the total variation of
their local conditional distributions (by the definition of
good and the compact-family bound (1.51)). So at a good vertex,
"disagreement" fails to occur with probability at least
q_0 = 1 - e^{-K eps} for a constant K. At a bad vertex,
disagreement is certain.
Along a path of r consecutive coarse vertices,
Pr( no disagreement along the path )
<= prod_{X} ( q_0 * 1_{X good} + zeta(b) * 1_{X bad} + zeta(b) )
<= ( q_0 + 2 zeta(b) )^r, (1.56)
where the last bound uses (1.55). Choose eps small enough that
q_0 < 1/4 and then b large enough that zeta(b) < 1/4 (this is
possible because eps is fixed before b in the order of choices).
Then the per-step factor is at most 1/2, so
Pr( disagreement along the path ) >= 1 - 2^{-r}. (1.57)
DECORRELATION. Along a path of coarse cells from x to y, the number
of coarse steps is r = ceil( d / s ). Under the exploration
coupling,
E_mu[ O(q_x) O'(q_y) ]
= E[ O(q_x^1) O'(q_y^2) ]
= E[ O(q_x^1) O'(q_y^2) | couple at y ] Pr( couple at y )
+ E[ O(q_x^1) O'(q_y^2) | disagree at y ] Pr( disagree at y ).
When the copies couple at y, the pair (q_x^1, q_y^2) has a joint law
in which q_x^1 is Haar-uniform on S^2 (because g_x^1 is Haar-
uniform, and O is a function of q_x^1 which has mean zero against
Haar measure); hence E[ O(q_x^1) | couple ] = 0. So the coupled
term vanishes identically.
Therefore
| E_mu[ O(q_x) O'(q_y) ] |
<= ||O||_infinity ||O'||_infinity Pr( disagree at y )
<= ||O||_infinity ||O'||_infinity 2^{-ceil(d/s)}. (1.58)
Absorbing 2^{-ceil(d/s)} <= 2 * 2^{-d/s} into the constant,
(1.52) follows. []
Extension to F, G locally Lipschitz. Any real bounded Lipschitz F
supported at A is a sum of one-site observables plus a constant,
and the constant drops from Cov. The covariance of F and G is a
sum of at most |A| * |A'| two-point covariances, each bounded
by (1.58). The polynomial factor in |A| + |A'| + d is the number of
such pairs times the growth of the Lipschitz constant under the
sum decomposition. This gives (1.2) with the stated polynomial
p > 0 and c > 0.
Explicit scale X(b). The one-site bound applies at coarse scale
s where s = R_* b^{D + C_0}. Combining with Lemma 1.4 at h = b
(giving log R_* <= 4 pi b + C_D sqrt( B log B ) with B = b) and
setting R = 1 gives
log s(b) <= 4 pi b + (D + C_0) log b + C_D sqrt( b log b ). (1.59)
Fix eta in (0, pi). For all b >= b_0 (with b_0 chosen large
depending on D, C_0, C_D, eta), the three contributions
( D + C_0 ) log b and C_D sqrt( b log b ) and C_D on the RHS are
absorbed by eta * b. Enlarge the constant X(b) accordingly, giving
X(b) = C exp( (4 pi - eta) b ). (1.60)
Uniqueness of the infinite-volume limit. Under (1.2), the
Cov( F, G ) between any two compactly supported local observables
is at most exponentially small in the lattice distance. All local
expectations of the form E_mu_{Lambda}[ g ] for fixed g therefore
form a Cauchy sequence in the volume Lambda -> Z^2 limit. The
limit is unique because two cluster points differ by at most the
exponential factor from (1.2). This proves existence and
uniqueness of the periodic infinite-plane state mu_beta for every
beta >= b_0.
The remaining assertion of Proposition 1.1 is the rectangular
doubling estimate (1.3). This is a strictly positive-definiteness
statement about the rectangular transfer operator on a finite
torus. Its proof is more delicate than the covariance bound; it
requires the auxiliary trace test of the trace estimates, not the one-site
decorrelation above.
The two assertions (i) and (ii) are established in the original
proof by the same stiffness decrease, with (ii) obtained by
applying a rectangular transfer operator argument to the trace
of an appropriately chosen minorant. Item (ii) is deferred to the trace estimates,
where the necessary transfer operator and rectangular partition
function Z_beta(n, w) are introduced. This completes the preliminary correlation-length estimate
for the purposes of the present construction: (1.2) is the bound used
in the block observation-5, and (1.3) is used only in the trace estimates.
Here is every Section of the proof in one sentence, without algebra.
This is the fastest way to see what the paper actually does.
the preliminary correlation-length estimate. Above a certain length scale X(beta) the model
decorrelates exponentially. X(beta) is enormous (its logarithm is
essentially 4 pi beta, matching the Kosterlitz-Thouless mass
formula) but finite and explicit. The proof uses a stiffness argument
based on the O(3) Ward identity.
the block observation. Set up the block renormalization. Partition the lattice
into L-by-L blocks. Each block gets a retained spin obtained via a
fixed averaging kernel. The kernel and its covariance matrix are
written down exactly; one slightly rotated (inclined) version is
also constructed, needed later to prove rotation invariance.
the exact renormalization step. Each exact block integration leaves an expression of the
same form up to: (a) a coarse spin for the retained field; (b) a
small "canonical" correction given by a finite polynomial in the
retained spin; (c) a tiny "cluster gas" correction whose amplitude
is exponentially small in a scale p_j. Each integration step
contracts corrections by a factor at least L^{2 - 2 upsilon} with
upsilon small.
the integrated comparison. Cluster sums are compared with their leading
approximation, with absolute error bounds that are exponentially
small in the number of active sites.
the shooting construction. Choose the bare coupling so that after N integration
steps the renormalized coupling equals a fixed reference value H
exactly. Done for every large N by a two-endpoint continuity
argument, called "shooting". Result: for every depth N, there is an
admitted bare coupling beta_N.
the trace estimates. Use two transfer-operator tests to prove mass. The
rectangular doubling test shows that iterating the transfer operator
over doubling rectangles multiplies the excited-trace by a factor
strictly below 1. The eight-cell alignment test shows that the
limiting distribution of eight specific coarse cells is not Gaussian.
the renormalization of the sources. Sources. Carry linear sources z_x . q_x through every
integration step. The linear response coefficient c_j is positive
and computed by a convergent sum. The accumulated field
normalization B_N = prod_{j=1}^N c_j^{-1} is shown to be positive,
to converge as N -> infinity, and to agree with the canonical
normalization 1 / (xi_beta sqrt(chi_beta)).
the continuum and infinite-volume limits. Continuum limits. The field moments are bounded
uniformly in the lattice spacing and in the depth. This implies
factorial tails, which in turn imply that the generating functions
converge in a complex polydisc and hence determine the limiting
distribution uniquely.
the reconstruction. Osterwalder-Schrader reconstruction. Check that the six
Euclidean axioms (A1)-(A6) hold for the limiting
hierarchy. Reflection positivity comes from tiling the half-space
with reflection-symmetric pairs of slabs. Clustering comes from
the trace estimates. The mass gap comes from the trace test in the trace estimates.
Non-Gaussianity comes from the eight-cell alignment test.
The trajectory decomposition and the volume estimates. Every beta -> infinity, not only the specific ones
constructed in the shooting construction. Write beta = H + gamma N + s with N
integer and |s| bounded. Show that every such trajectory is
admitted (the "confinement" of trajectories), and that the
resulting small discrepancies -- the difference between two
nearby trajectories -- are uniformly bounded in terms of |s - s'|.
the fixed-offset field construction. Show that the second-moment length and susceptibility
are finite and positive for every bounded offset s, and that the
canonically normalized fields converge: on the plane, on any fixed
torus, in law and in every joint moment.
the uniqueness argument. Uniqueness after canonical normalization. Compare the
ordinary trajectory at offset s with (i) the same physical coupling
viewed at depth N+1, whose offset is s - gamma, and (ii) the same
physical coupling viewed through the inclined first blocking, whose
offset is s + d, where d is irrational. These two comparisons are
the "dense-periods" or "two-scales" identities. Together with Haar
translation and continuity they imply the limit does not depend on s
or on N. That completes Theorem B.
The plain-English message: the O(3) model in two dimensions is hard
because the coupling is marginal at the Gaussian fixed point (its
tree-level beta function vanishes). The proof handles this by
working with the canonical normalization directly: instead of
requiring that beta_N has a specific limit, it uses the running of
the coupling under exact blocking to define field normalization and
to identify the limit unambiguously. The massive nature of the
resulting theory is a consequence of the Kosterlitz-Thouless
mechanism, made rigorous via the reflection-positive transfer
operator.
Introduce the block averaging that drives the exact RG
step. State the exact cell geometry, the retained-spin observation,
and the three free-kernel estimates used in the exact renormalization step. This is the
only place where free covariance information enters the proof.
Fix a dyadic integer L >= 2. Partition Z^2 into ordinary blocks
B_Y = { x in Z^2 : Y*L <= x < (Y+1)*L }, Y in Z^2. (2.1)
Each block has L^2 fine sites. The coarse lattice is L Z^2.
Inclined cells (needed for the first step only). Let
O_+ = (1/5) [[3, -4], [4, 3]], det O_+ = 1, (2.2)
a rational rotation of Pell type (order 5 in SO(2) of the
denominator-cleared integer lattice). Put L_* = 5L. The inclined
cells are the fundamental domains of the lattice generated by the
block-translation vectors (3L, 4L) and (-4L, 3L), which has
determinant 25 L^2 = L_*^2.
LEMMA 2.1 (Inclined geometry). The inclined cells B_Y^{O+}
partition Z^2, each containing exactly L_*^2 = 25 L^2 fine sites.
Each cell is connected by nearest-neighbour paths. No fine site
lies on a cell boundary. Quarter turns about a cell center preserve
the partition (up to relabelling), as does reflection in a
coordinate axis. Every site of a cell can be joined to the central
region of that cell by a within-cell path of length O(L_*).
Proof. The two block-translation vectors are integral with
determinant L_*^2 = 25 L^2. Suppose for contradiction a boundary
site y is also a fine site of Z^2. The boundary of an inclined cell
passes through translations of the shifted vector by an integral
combination of (3L, 4L) and (-4L, 3L). Since the entries 3 and 4 are
odd while L and 3L - 4L = -L are odd multiples, adding any nonzero
lattice vector either changes one coordinate parity or contributes a
half-integral shift; in every case a fine site of Z^2 cannot coincide
with a boundary point. Each half-open cell is therefore a
fundamental region of the translation lattice, and its fine sites
form a complete set of coset representatives. The index of the
lattice equals 25L^2. Quarter turn and reflection follow because
O_+ rotates the block-translation lattice onto itself (up to a
small integral shift of the center at (1/2,1/2)) and preserves Z^2;
the product weights w_Y transform accordingly. Connectivity and
path length are elementary properties of the rotated square of side
L_* inside the fine lattice. []
For each cell Y (ordinary or inclined) choose a real weight function
w_Y : Z^2 -> [0, infinity), (2.3)
supported on Y (or a fixed small dilation of Y), with
sum_x w_Y(x) = 1, symmetric under the cell symmetry group,
bounded: sup |w_Y| <= C/L^2 (ordinary) or C/L_*^2 (inclined),
with discrete first and second differences bounded by C/L^3 or
C/L_*^3, and vanishing to order O(exp(-cL)) outside a collar of
width L/4 (or L_*/4).
The retained mean vector at Y is
m_Y = sum_{x in Z^2} w_Y(x) q_x in R^3. (2.4)
The retained spin V_Y is m_Y / |m_Y| (normalized-mean branch),
provided |m_Y| >= 1/2. If |m_Y| < 1/2 (fallback branch), one uses a
measurable selection m_Y^hat with |m_Y^hat| = 1; the choice does
not affect any estimate.
The chord process. For a bond e = {x, y}, set r_e = |q_x - q_y| in
[0, 2]. From sum_x w_Y(x) = 1 and |q_x| = 1 for all x,
1 - |m_Y|^2 = (1/2) sum_{x, y} w_Y(x) w_Y(y) |q_x - q_y|^2. (2.5)
This exact identity is used in every mean-error estimate below.
LEMMA 2.2 (Mean error). Uniformly in the cell Y (ordinary or
inclined), the pin configuration, the strengths beta_e in [0, b],
and the spin configuration q,
| m_Y - m_Y^hat | <= (C/L) sum_{e in B_Y} r_e, (2.6)
| m_Y - m_Y^hat | <= (C/L_*) sum_{e in B_Y^{O+}} r_e, (2.7)
with an absolute constant C > 0. If every nearest-neighbour chord
in the cell satisfies r_e <= T for some T with C L_*^2 T^2 <= 3/4,
then the fallback branch is not taken and
| m_Y - m_Y^hat | <= 1 - |m_Y|^2 <= C_L T^2. (2.8)
Proof. For (2.6): join any pair x, y in the cell by a coordinate-
aligned lattice path inside the cell (length O(L) ordinary, O(L_*)
inclined). In the weighted sum (2.5), each coordinate step in row s
of the path is loaded by the row's total weight, sum_{x in row s}
w_Y(x) <= C/L (or C/L_*); the discrete loadings add up to a
geometric bound. Summing over all pairs x, y in B_Y and using the
weighted loadings,
sum_{x, y} w_Y(x) w_Y(y) |q_x - q_y|
<= (C/L) sum_{e in B_Y} r_e.
Combining with (2.5) gives
1 - |m_Y|^2 <= (C/L) sum_{e in B_Y} r_e. (2.9)
On the normalized-mean branch, |m_Y - m_Y^hat| = 1 - |m_Y| <=
1 - |m_Y|^2, giving (2.6). On the fallback branch |m_Y| < 1/2,
every projection of m_Y onto S^2 satisfies |m_Y - q_x| <= 1 + |m_Y|
<= 2(1 - |m_Y|^2), giving (2.6) again after possibly enlarging C.
For (2.7) the identical argument with L_* replacing L applies.
For (2.8): if r_e <= T everywhere, then by (2.5),
1 - |m_Y|^2 <= (1/2) sum_{x, y} w_Y(x) w_Y(y) (C L_* T)^2
<= C L_*^2 T^2.
Choosing T with C L_*^2 T^2 <= 3/4 yields |m_Y| >= 1/2 so the
fallback branch is not taken, and |m_Y - m_Y^hat| = 1 - |m_Y| <=
1 - |m_Y|^2 <= C_L T^2. []
COROLLARY 2.3 (Small-chord mean). At the terminal scale of a layer
j, with r_e <= T_j = H_j^{-1/2} p_j throughout the cell, the mean
error obeys
| m_Y - m_Y^hat | <= C_L T_j^2 (2.10)
uniformly in Y and in the layer j.
After linearizing the exact integration around a slowly varying
reference spin, the Gaussian factor in the linearization is a free
covariance matrix. Write K_h for the coarse-kernel covariance
associated with a history h of h blocking steps. K_h(k) is a real
symmetric 3-by-3 matrix per coarse momentum k in [-pi,pi]^2; for
ordinary blocks K_h = K_h^{scalar} Id_{R^3}, and for inclined
starts K_h differs by a small rotation.
LEMMA 2.4 (Principal log, Fourier bounds). There is a constant
delta_0 = delta_0(L) > 0 such that for every history h, K_h(k)
extends analytically to the strip
Omega_{delta_0} = { p in C^2 : |Re p_mu| <= pi + delta_0,
|Im p_mu| <= delta_0 }, (2.11)
and satisfies, on that strip,
K_h(p) >= c_0 * Id_{R^3}, (2.12)
with c_0 = c_0(L, delta_0) > 0 uniform in h.
Proof. On the coarse Fourier lattice K_h(k) is a sum over fine
aliases of |b_Q(k_alpha)|^2 / D(k_alpha) times a smooth reference
amplitude, where b_Q is the Fourier amplitude of the block
averaging kernel and D is the free mass symbol D(p) = 4 sum_mu
sin^2(p_mu / 2). Non-principal aliases |alpha| >= 1 satisfy
|D(k_alpha)| >= c(1+|alpha|)^2; the centred bump decays as
(1 + |alpha|)^{-J} for any fixed J. The resulting series converges
uniformly on the strip and gives (2.11). The lower bound (2.12)
follows from ellipticity of the principal alias together with the
strip bound on the free symbol. All constants are fixed once L is
chosen and depend only on L and delta_0. []
LEMMA 2.5 (History stability). Let h_1, h_2 be two histories sharing
their last r >= 0 ordinary observation steps, with the same kernel
w_0 and identified output coordinates. Then for every delta_1 <
delta_0,
|| K_{h_1} - K_{h_2} ||_{an, delta_1}
<= C ( A_0 / L^2 )^r, (2.13)
where || . ||_{an} is the sup norm on Omega_{delta_1}, A_0 > 0 is a
fixed constant (the local multiplier in the free propagation), and
C depends only on L and delta_1. The same bound holds for any
prescribed finite set of analytic derivative and exponential kernel
norms after decreasing delta_1 or enlarging the weight. When h_1,
h_2 are followed by a common next ordinary observation, their P_h
and C_h differences obey (2.13) with a permitted extra factor C_L.
Proof. Each ordinary observation step is a coarse averaging combined
with a fine isometry. The averaging lowers non-principal
frequencies by a factor A_0/L^2; the principal alias is transported
by an isometry. After r shared steps, K_{h_1} - K_{h_2} is a product
of r contractions applied to the low-frequency mismatch. The
contractions contribute the factor (A_0/L^2)^r; the mismatch is
uniformly bounded. The "common next observation" clause uses the
same argument followed by a bounded common map. []
The inclined first block reuses all of the above with L_* replacing
L. The averaging kernel w^{O+} is defined by composition with
O_+^{-1}; the reflected version w^{O-} is the reflection of w^{O+}.
Lemmas 2.1-2.5 carry over verbatim with L_*, giving the
following four facts:
(i) Mean error (2.6)-(2.8) holds for the inclined geometry with
L_*.
(ii) Principal Fourier lower bound (2.12) holds for the inclined
K_h^{O+}.
(iii) History stability (2.13) holds for either inclination from
the second step onwards.
(iv) After the inclined first step, subsequent ordinary steps are
identical to those of the ordinary trajectory.
Fact (iv) is what makes the inclined and ordinary trajectories
comparable in the uniqueness argument: they share every step beyond the first.
Everything in the exact renormalization step-13 uses exactly:
(a) Lemma 2.2 (mean error) in both geometries;
(b) Corollary 2.3 (small-chord mean);
(c) Lemma 2.4 (principal Fourier log);
(d) Lemma 2.5 (history stability).
No other property of the free kernels is used at any later point.
Prove that integrating all interior fine spins in each
block, starting from any admitted retained density at layer j, gives
back the SAME FORM of retained density at layer j-1, up to:
(a) a precise coupling recursion b' = b + alpha + kappa/b + Delta;
(b) an updated canonical finite polynomial P' = C_h(P);
(c) a regular error f' with |f'| <= q |f| + tiny, q < L^{-2+upsilon};
(d) a covering sum Xi' with cap w_{j-1}.
The step is EXACT (no approximation in the "sector" handling); the
five "stages" are the bookkeeping that isolates each kind of term.
Retained class at layer j. A retained density at layer j with
precise coupling b and canonical list P is a probability measure on
V-configurations on the coarse torus Z_j = (Z/L^j Z)^2 that is
absolutely continuous with respect to prod_Y dsigma(V_Y) with
density Z_j^{-1} times
exp( b sum_{<Y,Y'>} V_Y . V_Y' [precise kinetic]
+ F_P(V) [canonical bulk polynomial]
+ f(V) [regular error]
+ log Xi(V) [mandatory covering sum]
+ source terms (see 7) ), (3.1)
where:
(i) b in [H_j/2, 2H_j], (admission)
(ii) P = (P_4, P_6, J_2, I_2, I_4) is a finite list of bulk
canonical coefficients, defined below in 3.4;
(iii) f is a regular error: it is a finite sum of smooth local
functions supported in graph domains, each carrying an
exponential localization (load) s_X >= 1, and it satisfies
|f|_{k,j,A} = sup_o sum_{X: anchored at o} e^{A s_X} [f_X]_{k,j}
<= delta_j = H_j^{-2.05}, (3.2)
for k = 8 derivatives and an absolute support constant A;
(iv) Xi is a mandatory covering sum: a finite sum over connected
site animals of configurations bad bonds, with weights
bounded by cap activities k_lambda satisfying
sup_x sum lambda e^{2 s_lambda} |k_lambda| <= w_j =
exp(-p_j^{1/4});
(v) the source terms are as in the renormalization of the sources.
THEOREM 3.1 (Exact step and comparison). Let upsilon in (0, 1/4) be
fixed. There are choices of L, of the canonical caps and comparison
weights, of P_0, and finally of H, such that integration over one
block observation sends every admitted retained density at layer j
and b in [H_j/2, 2H_j] to an EXACT retained density of the same
form at layer j-1, with all renewed caps. Its precise coupling
satisfies
b' = b + alpha_h(P) + kappa_h(P) / b + Delta,
|Delta| <= C_L H_j^{-1.05}, (3.3)
where alpha_h, kappa_h are fixed polynomial functions of P and of
the free history h (the canonical extraction).
For two regular steps with common geometric scales, write
lambda = b_2 - b_1 and let u be the fixed weighted discrepancy
(canonical coefficient differences + regular-error difference /
delta_j + covering-gas difference / w_j). Then
u' <= q u + C_L poly(H_j) epsilon_h
+ C_L (log H_j)^C |lambda| / H_j, (3.4)
|lambda' - lambda| <= C_L u + C_L poly(H_j) epsilon_h
+ C_L (log H_j)^C |lambda| / H_j,
q < L^{-2 + upsilon}. (3.5)
Here epsilon_h = 0 for identical histories; epsilon_h <=
(A_0/L^2)^r for two histories that share their last r ordinary
observation steps (Lemma 2.5).
The contraction (3.4)-(3.5) with q < L^{-2+upsilon} is the
fundamental statement of the whole construction. Its meaning: each
exact blocking step SHRINKS discrepancies between two nearby
inputs by a factor at least L^2 (up to a small slack). The factor
L^2 equals the "marginal dimension" of 2 in two dimensions; its
positivity is what makes the recursion work.
Before integrating, split the fine spins by chord size. Fix t_j
from the layer-j scale.
Definition. A bond e = {x,y} is a true bad bond if r_e = |q_x -
q_y| > t_j. Let f(x) = max_{e : x in e} r_e. Choose eps > 0 small
and set a = eps t_j.
Mark. Mark every site of every integral square box of radius at
most 1/4 L^0 (a fixed lattice unit) on which the average of f
exceeds a.
Resolve. Let U be the union of marked boxes, dilate U by
d = C D_1 with D_1 ~ (log H_j)^2, take the max-distance connected
components, and retain those components containing at least one
true bad bond. Call them E_C.
The retained components are the "sectors" in which the fine spins
fail to be small perturbations of their reference. They satisfy:
- distinct retained components have non-adjacent sites;
- every true bad bond lies strictly inside exactly one retained
component;
- outside all retained components, all chords are <= t_j.
The point of the sector decomposition is: inside each retained
component the fine spins are NOT small, so no expansion is applied;
outside them, all chords are <= t_j, so the harmonic linearization
is valid. The two regimes combine as
measure = (exact density on sectors) * (linearized density outside).
Outside the sectors, linearize around a reference spin e_0 in S^2.
Write q_x = Exp_{e_0}(xi_x) with xi_x in the tangent plane at e_0.
For small |xi|,
q_x . q_y = 1 - (1/2) |xi_x - xi_y|^2
+ (1/24)( |xi_x|^4 + |xi_y|^4 - 6 |xi_x|^2 |xi_y|^2 ) ... (3.6)
Since q_x . q_y = b cos of the relative chord, the quadratic part of
the kinetic energy is
b sum_e (q_x . q_y - 1) = -(b/2) sum_e |xi_x - xi_y|^2 + O(xi^4). (3.7)
The exact block weight for the small-angle sector is the exponential
of (3.7) plus a smooth remainder R(xi) = O(xi^4) and higher orders.
Gaussian completion. Introduce a single ambient Gaussian Z ~
N(0, C_s tensor Id_{R^3}) with C_s = (1/2)(Pi_0 + Pi_0^*) the
free covariance matrix of Lemma 2.4, and substitute xi = mu + g Z
with g = b^{-1/2}, mu the conditional mean. Expanding the
quadratic in Z and using C_s^{-1} - A = (AC_s - Id)C_s A + ...,
exp( (b/2) sum_e q_x q_y - b/2 * const )
is written as the Gaussian density in Z times an explicit
density-ratio factor expr( (1/2) Z^* (C_s^{-1} - A) Z
- g^{-1} (A mu - dinv) Z + ... ),
where the residual terms are summable in |Z| because Pi_0 is
elliptic and the residual E_s = A C_s - Id is exponentially small.
The resulting integral is therefore
Gaussian_mean_{Z ~ N(0, C_s)}( density_ratio(Z) ),
with density_ratio an explicit analytic function of Z. This is
the "ambient Gaussian representation" of the free block integration;
it uses the SAME free covariance matrix in every history h, so the
constants depend on L but not on j.
The Taylor expansion of the density-ratio factor in Z, ordered by
field degree and mesoscopic order, is the "canonical" content of
the step. The extraction pinpoints the finite polynomial part as
new canonical coefficients, and leaves the rest as a regular error.
The field variables after the Gaussian average. Write (see Section
3 of the source) the reference spin as V_Y(s), let (mu_X)_Y denote
the conditional mean of the fine spins given the boundary data, and
let f_Y = m_Y - mu_X (the "deviation"). The Taylor expansion of
the block Gibbs measure in the field deviation, ordered by total
field degree r and mesoscopic suppression g^{2d}, is
sum_{r,d} F_{r,d}( V, backgnd ), with F_{r,d} homogeneous of
field degree r + 2d and r in {0, 1, 2, 3, 4}.
Triangular canonical prescription. Define, using the free block
symbol K' on the next layer (the output kinetic form):
P'_4 = F_{2,1},
alpha = affine Hessian of F_{2,0},
I'_2 = Comp( F_{2,0} - alpha S_{K'}^{[2]} ), (3.8)
P'_5 = 0, I'_3 = 0,
P'_6 = F_{4,1},
I'_4 = F_{4,0} - alpha P'_4 - alpha S_{K'}^{[4]},
kappa = affine Hessian of F_{4,-1},
J'_2 = Comp( F_{4,-1} - kappa S_{K'}^{[2]} ). (3.9)
Here:
- S_{K'}^{[n]} is the homogeneous degree-n part of the output
kinetic energy, assigned to its anchor by splitting edge terms
between their endpoints;
- the "affine Hessian" is the bulk sum per anchor of the second
derivative in the affine test V_Y -> cos(alpha V_Y z_B) at
alignment;
- Comp(...) is the quadratic-packet compensation of the exact renormalization step of
the source: it removes any residual affine Hessian and returns
the packet to the standard form;
- the odd slots P_5, I_3 are zero because the S^2 target has no
cubic invariants under O(3) x spatial inversion.
These formulas define the CONICAL MAP
P' = C_h( P )
depending only on the free history h and incoming canonical
coefficients P. It is a fixed finite algebraic operation. It does
not depend on the precise coupling b, the cutoff, the regular error
f, the covering sum Xi, or the period.
The precise-coupling recursion is then
b' = b + alpha_h(P) + kappa_h(P) / b + Delta, (3.10)
with Delta the sum of: (i) affine-Hessian extractions not captured
in alpha, and (ii) the tiny O(L H_j^{-1.05}) residual.
Precise-coupling constants. At H_j large, alpha_h and kappa_h are
precomputable in the limit h -> infinity (they converge; see
the shooting construction). Their limits are the Kosterlitz-Thouless increments
alpha_infinity -> -gamma = -log L / (2 pi),
kappa_infinity / b -> --- (of order 1) * (1 / b).
The first is negative: each step DECREASES the coupling by
(log L)/(2 pi). This is the asymptotically-free running of the O(3)
model, matched to the marginal coupling b ~ H_j + gamma j.
The remaining correction after canonical extraction is a regular
error: an explicit function of the field, at higher Gaussian order
and higher Taylor order. Its size is controlled by:
LEMMA 3.2 (Regular-error contraction). For every fixed required
output support exponent chosen before L,
| (T f) |_{k,j-1,B} <= ( C/L^2 + o_H(1) ) | f |_{k,j,A}, (3.11)
where f is any old regular error and T is the "isolated response"
operator
(T f)(V) = E_{C_s} ( chi_X(q) f(q) ), (3.12)
q_x = Exp_{T_x}( proj_{T_x-perp}(mu_x + g Z_x) ),
with chi_X a prepared smooth graph cutoff. The little-oh is uniform
in H_j >= H. The same bound holds for an input DIFFERENCE through
order k-1. A map change acting on an unchanged input satisfies the
original map-difference estimate of the assembly of the exact step with its polynomial
losses and one extra derivative on that input.
Proof. The key point is the L^{-2} coefficient, computed as follows.
Write q_x = predicted_x + (L^{-1} mismatch) + (L^{-2} mismatch) +
(L^{-3} + o_H(1)) mismatch, where predicted_x is the affine mean
prediction at x. The exact Taylor expansion (the Gaussian completion) gives the
bounds on the three tangent sectors:
- one-linear = C/L,
- one-quadratic = C/L^2,
- remainder = C/L^3 + o_H(1).
Nonlinear chart terms and window truncation belong to the
remainder. The quadratic term is the most delicate: its two
linear-space predictions vanish (by affine normalization of the
canonical list P), and the linear-quadratic predictions cancel
under position inversion x -> 2 o - x of the packet (again by
affine normalization). So only the (C/L^2) coefficient survives.
The output anchor removes the "constant" term by weighting; the
first tangent term is exactly the conditional-mean prediction, so
it is subtracted. The (1/L^2) coefficient is the residual
"first-quadratic" Taylor term after these subtractions. Summing
over the L^2 old anchors and the output-support weighting gives
(3.11).
Higher Taylor orders (degree >= 4) give an additional g factor per
degree: total contribution g^{4.6} at 4th order, g^{4.5} at
higher; these are absorbed by the spare margin in (3.4)-(3.5),
because q < L^{-2 + upsilon} strictly for upsilon small.
Exact S^2 subtleties. Internal O(3) invariance removes the
odd-order (cubic, quintic) terms of the analytic factor. The
"spherical Jacobian" of the measure is opened as a smoothly
cut-off ratio J(u)/J(0) - 1 starting quadratically with order g^2.
The ambient Gaussian has three colors, of which one is literally
cancelled by the scalar normalization; the remaining determinant
coefficient 2 is used in all estimates. Tangent inversion in the
anchor stabilizer (a subgroup of O(2) = O(3)/O(2) x O(2)) is the
symmetry that zeroes the cubic Taylor term.
Combine:
(i) Sector decomposition (3.2) gives exact representation.
(ii) Gaussian completion (3.3) gives ambient Gaussian integral.
(iii) Canonical extraction (3.4) gives P' = C_h(P) with exact
coefficients and precise-coupling recursion b' = b + alpha
+ kappa/b + Delta.
(iv) Regular error contraction (3.11) gives new error cap
delta_{j-1} = H_{j-1}^{-2.05} + (C/L^2 + o_H(1))
delta_j.
(v) Covering sum Xi renews to activity cap w_{j-1} because
the sector interiors are handled without expansion, and
each retained region has an activity bounded by
H_j^{-2.05} times the number of retained regions (which
is polynomial in H_j).
For two nearby inputs (b_1, P_1, f_1, Xi_1) and (b_2, P_2, f_2,
Xi_2), write lambda = b_2 - b_1 and u as in the statement. Each
stage contributes a discrepancy of the stated size; combining:
- Stage (ii): Gaussian mean differs by O(u) + O(|lambda|/H_j).
- Stage (iii): canonical extraction is Lipschitz in the input
coefficients with constant C_L poly(H_j); affine correction
contributes O(|lambda|/H_j).
- Stage (iv): regular error of P' vs P'_1 differs by O(u) +
O(|lambda|/H_j) + tiny.
- Stage (v): covering cap discrepancy is absorbed by the strict
margin from w_j.
The specific form of the discrepancy (3.4)-(3.5) results, with the
contraction coefficient q < L^{-2 + upsilon} from (3.11). The size
restriction log(2+S) <= 16 B holds throughout because all sides are
bounded by s <= L^j R_0 <= exp(O(poly(log H_j))) << exp(16 B) for
B >= H.
This completes the proof of Theorem 3.1.
Two slightly different blockings of the same physical model
produce slightly different covering sums. The integrated L^1
difference is bounded by the sum of the small individual label
discrepancies, not by the full partition function. This is what makes
the shooting argument (the shooting construction) and the fixed-layer matching
argument (the last four parts of the proof) work.
Fix a large H. On a terminal axis torus, let e^X tXi and e^X Xi be
two standard endpoint densities, in the retained-density class of
the retained-density class, with the SAME kinetic energy X (the b-sum_e q_x . q_y
term plus the O(3) target structure) and with two distinct covering
sums tXi, Xi, each of the form
Xi(q) = sum over compatible families Gamma of
prod_{lambda in Gamma} k_lambda(q), (4.1)
with each label lambda carrying:
- an inventory J_lambda of bonds (all bad bonds in the affected
sites -- mandatory covering; empty inventory contributes 1);
- a site support P_lambda (all sites touched by the label);
- a load s_lambda >= 1 (the size of the support as a multiple
of |J_lambda|) with |P_lambda| <= R s_lambda for an absolute R;
- a signed or complex weight k_lambda(q) in absolute value at most
p_lambda (the activity envelope).
Compatible families have disjoint inventories (their unions cover the
bad-bond set) and pairwise disjoint supports.
Discrepancies. Define for each label lambda:
d_lambda = || tk_lambda - k_lambda ||_infinity, (4.2)
p_lambda = || k_lambda ||_infinity + || tk_lambda ||_infinity. (4.3)
Covering-cap assumption. There is an absolute constant
Cap0 > 0 (determined by L, P_0, and the fixed support constant A)
such that
sup_x sum_{lambda : x in P_lambda} e^{A s_lambda} p_lambda <= Cap0. (4.4)
In the shotting and trajectory applications, the constant Cap0 is
taken INCREDIBLY small (of order w_j, exp(-p_j^{1/4})), so that all
the finiteness arguments below are enormously uniform.
LEMMA 4.1 (Energy decomposition). Fix D_1 an integer comparable with
(log H)^2. In the kinetic energy X, truncate the exponentially
localized free kernel at radius D_1, and group:
- the resulting positive square from the harmonic projection
difference, at each bond e;
- the direct HS_t term at the same bond,
into a single nonnegative quantity Y_e reading a fixed stencil of
size O(D_1). Then on a terminal torus of volume v:
X = - sum_e Y_e + X_rem, (4.5)
| X_rem | <= C_L v, (4.6)
| X_rem(q) - X_rem(q') | <= C_L n for q, q' differing
at n sites, (4.7)
Z >= e^{-C_L v log H}, (4.8)
| Xi(q) | bounded by the number of |P_Gamma|: for each q,
| Xi(q) | <= C_L^{|bad set|} e^{number of active sites}. (4.9)
Proof. Step 1 (kinetic decomposition). At each bond e = (x, y) write
the chord q_x - q_y in harmonic and anti-harmonic parts:
q_x - q_y = P_H(q_x - q_y) + (1 - P_H)(q_x - q_y),
where P_H is the harmonic (spatially smooth) projection of
the sector decomposition. Expanding the square,
(b/2)|q_x - q_y|^2 = (b/2)|P_H(q_x - q_y)|^2 + HS_t(q_x - q_y),
where HS_t is the truncated Hessian-of-tangent form of the canonical extraction
directly attached to the anti-harmonic content. Both terms are
nonnegative (the harmonic square trivially, HS_t by the convexity
of the sigma-model action). The free (harmonic) part is the one
playing the role of the Gaussian kinetic energy in the block
integration.
Step 2 (truncation). The kernel governing the harmonic square is
the free block kernel K_h of the block observation, which decays exponentially
in the lattice distance at rate exp(-c |x| / (log H)^2). Truncating
at radius D_1 = O((log H)^2) introduces
|truncation error| <= C_L exp(-c D_1) + O(chord^2)
per site, absorbed in X_rem. Thus
X = - sum_{e in Z^2} Y_e + X_rem,
with Y_e := (b/2)|P_H(q_x - q_y)|^2 + HS_t(q_x - q_y) the truncated
direct energy.
Step 3 (properties of X_rem). X_rem is a sum of kernel tails (each
bounded by C_L exp(-c D_1)) and Taylor corrections from the
truncation. Uniformly in the configuration,
|X_rem| <= C_L v.
Each term of X_rem is Lipschitz in the spin under dsigma-metric,
with bounded derivative, so for q, q' differing at n sites,
|X_rem(q) - X_rem(q')| <= C_L n.
Step 4 (lower bound on Z). Restrict the integral defining Z to the
neighborhood
Cap_v := { q : |q_x - q_y| <= c' for every neighbouring bond (x,y) }.
On Cap_v, HS_t(q) = 0 at every bond and the harmonic square is
bounded by (b/2) c'^2 per bond. So the integrand e^{-sum_e Y_e} is
at least exp(-C_L c'^2 v). The normalized product-of-area measure
has Cap_v-mass at least (c'^2/4)^{v} (each vertex independently in
a spherical cap of radius c'). Hence
Z >= exp(-C_L c'^2 v) * exp(v log(c'^2/4)) >= exp(-C_L v log H),
with c' = H^{-C} for a fixed large C.
Step 5 (upper bound on |Xi|). The covering sum Xi is a finite sum
over connected animals on Z^2, with activities bounded by the
covering cap. The number of connected animals of size s with degree
<= 4 is at most C_0^s for a fixed C_0. Summing over s <= |bad set|
|Xi(q)| <= sum_s C_0^s e^{-c s} <= C_L^{|bad set|} * e^{C_L |bad set|},
which is the stated form. []
LEMMA 4.2 (Exponential moments of rows: chessboard). Let B_T be a
sufficiently large dyadic integer, comparable with a fixed power of
log H. On tori whose periods are multiples of 2 B_T, for every set I
of unoriented rows (horizontal or vertical lattice rows),
E_mu exp( theta sum_{e in I} Y_e )
<= exp( C_L |I| B_T^2 log H ), (4.10)
theta = 1 - C_L D_1 / B_T. (4.11)
In particular B_T may be chosen so that theta > 0 and 1 - theta is
smaller than any prescribed fixed inverse power of log H.
Proof. Tile the torus with B_T-by-B_T tiles. On each tile, the
retained seam reflection positivity of the preliminary correlation-length estimate (see also the
classical reflection positivity framework of Frohlich, Israel,
Lieb and Simon 1978, 1980) gives
E_mu exp( theta sum_{e in tile} Y_e ) <= exp( C_L B_T^2 log H ).
Averaging over all B_T^2 translations of the tile grid: every row of
the torus is retained in at least a fraction 1 - C_L D_1 / B_T of the
translations (this is where the truncation scale enters). Since the
Y_e are nonnegative, applying generalized Hölder (in row and column
directions) to the union of retained rows gives (4.10). []
Recall the retained components E_C of the sector decomposition: dilation-power
union of marked boxes containing at least one true bad bond.
LEMMA 4.3 (Editable regions). Fix eps > 0 small (independent of H),
a = eps t_0, and let d = C D_1 be the dilation. Then:
(a) Each editable region E_C is an "isolated component": distinct
regions E_C have max-neighbor distance at least 2d apart.
(b) Kinetic lower bound. In each region containing a true bad
bond:
sum_{e in E_C} Y_e >= c H t_0^2 |E_C| = c p_0^2 |E_C|. (4.12)
(c) Sampling bound. In regions where the goal is to fill interior
spins with a smoothed boundary extension, the local sampling
cost is at most C / t_0^2 <= H^C per site.
(d) Upper bound. Replacing any selected collection of regions
(each containing its true bad bonds) and using the smoothed
boundary extension:
1 - |m_Y|^2 + hs_t(q') <= c/2 p_0^2 |E_C| / d^2, (4.13)
where hs_t is the direct HS_t term.
Proof. Part (a) (isolation). By construction the dilation radius is
d = C D_1, and each retained component E_C is the closure of its own
dilated generating region: the union of marked boxes that contain
at least one true bad bond. Distinct marked-box clusters generate
disjoint retained components (each marked box belongs to one cluster
only), and their d-dilations remain max-distance at least d apart.
So every pair of distinct retained components E_C, E_C' has
max-distance >= d.
Part (b) (kinetic lower bound). On every marked box inside E_C the
average of the chord function f(x) := max_{e touching x} |q_x - q_y|
exceeds a = eps t_0. The generating true bad bond supplies at least
one adjacent pair with chord >= t_0. Splitting the local kinetic
energy into the direct quadratic HS_t term for chords below t_0 and
the positive linear branch of S_t for chords above t_0, then applying
Cauchy-Schwarz to the quadratic part, gives
sum_{e in E_C} local kinetic energy >= c H t_0^2 |E_C|,
with c = c(eps) a fixed positive constant once eps_0 is fixed. The
factor H t_0^2 = eps_0^2 p_0^2 is the per-site energy scale.
Part (c) (sampling cap). Fix q_0 in S^2 and let c t_0 be small. The
spherical cap Cap_t := { q in S^2 : |q - q_0| <= c t_0 } has area
pi (c t_0)^2 (1 + O(t_0^2)). Its normalized area is (c t_0)^2/4
(1 + O(t_0^2)). The normalized density of Cap_t is therefore at most
4/(c^2 t_0^2) (1 + O(t_0^2)) = C / t_0^2, which is the local
sampling bound needed by the fill.
Part (d) (returned chord control). The smoothed-extension construction
of the fill is C t_0-Lipschitz, so every chord inside the fill is
bounded by 2 C t_0 plus the chord of the boundary data. The factor c'
in the cap radius of Part (c) is chosen small enough that the
returned chords are all within C'' t_0 of zero. []
LEMMA 4.4 (Integrated gain per term). With notation as in Lemma 4.3,
let Q be the measurable set of original q-configurations "admitting"
a given edit term (defined before any sampling, by the mandatory-
cover inventory rules; the set Q is independent of the sampling).
Let fill denote the conditional distribution of interior spins in
the retained regions after the "good fill" is chosen. Then
(1/Z) int_Q e^{X(q)} E_{fill|q}[ Xi(q') ] dq
<= exp( - c p_0^2 n / d^2 ), (4.14)
with n = |E|, E the union of the region sites used in the term.
Proof. Using Lemma 4.1, X(q) - X(q') <= C n - c p_0^2 n / d^2 +
sum_{e in I_E} Y_e(q'). By Lemma 4.2 with theta = 1 - C_L D_1 / B_T,
replacing the last coefficient 1 by theta costs at most cp_0^2 n/(4d^2).
Then
X(q) - X(q') <= C_L n - c p_0^2 n / d^2
+ theta sum_{e in I_E} Y_e(q')
<= C_L n - (3c/4) p_0^2 n / d^2. (4.15)
The conditional distribution of q' given q has uniform bound
e^{C n log H}. Now integrate the original q over Q. By product
measure, the marginal of q on E is a probability; on the rest, the
estimate integrates trivially. Together:
integral <= exp( C_L n + C n log H - (3c/4) p_0^2 n / d^2 )
* E_mu exp( theta sum_{e in I_E} Y_e ).
With |I_E| <= C d^2 n, Lemma 4.2 gives
E_mu exp( theta sum_{e in I_E} Y_e ) <= exp( C_L d^2 n B_T^2 log H ),
absorbed by increasing C_L and the p_0^2 margin. This proves (4.14). []
PROPOSITION 4.5 (Integrated covering comparison). Under the covering-
cap condition (4.4) with a fixed constant Cap0 small enough, and
under the energy decomposition, exponential moments, and integrated
gain of the energy decomposition, one has
(1/Z) int e^X | tXi - Xi | dq
<= exp( sum_lambda d_lambda e^{2 s_lambda} ) - 1. (4.16)
The sum is over all label, and d_lambda = || tk_lambda - k_lambda ||_infinity
as in (4.2).
Sketch of proof.
(a) Forest representation. Write
tXi - Xi = sum over edit terms of (edit tree contribution),
edit term = a compatible family Gamma of labels, with at least
one label edited (weight tk_lambda - k_lambda instead
of k_lambda).
(b) Each edit term is a signed product over labels of (tk_lambda -
k_lambda) and (k_mu) for unedited labels. Its site support is the
union of P_lambda over the family.
(c) Bound each edit term by its absolute value:
| edit term | <= prod_{edited} | tk - k | * prod_{unedited} |k|
<= prod_{edited} d_lambda * prod_{unedited} p_mu.
(d) Since supports in the family are disjoint, integrate each label
independently using the integrated gain Lemma 4.4 for the union of
supports, and (4.10) for the exponential moments of rows inside.
The union of edit label discrepancies contributes
prod_{edited} d_lambda e^{s_lambda}.
The union of gas labels contributes
prod_{unedited} p_mu.
(e) Sum over edit trees via the tree-graph inequality of Penrose type
[FernandezProcacci]. The one-child sum is bounded by the covering
cap:
sup_x sum_{lambda : x in P_lambda} p_lambda e^{s_lambda} < 1/(16R),
for Cap0 small enough; the factor 1/16 gives strict room. Then
descendant trees have total weight at most e^{b/4} <= e^b for any
root of size b. In particular, the sum over tree shapes converges
and telescopes the product of edit discrepancies.
(f) The result is the right side of (4.16). []
(a) Lemma 4.1 (energy decomposition): X = - sum Y_e + X_rem with
remainder Lipschitz and controlled.
(b) Lemma 4.2 (row moment): exponential moment of sum Y_e on any
set of rows.
(c) Lemma 4.3 (editable regions): kinetic lower bound, sampling
bound on each retained component.
(d) Lemma 4.4 (integrated gain per term): every edit term costs
at least exp(-c p_0^2 n / d^2) in the integrated L^1 sense.
(e) Proposition 4.5 (integrated comparison): L^1 difference of
covering sums <= exp(sum d_lambda e^{2 s_lambda}) - 1.
These are used:
- in the shooting construction, via the L^1 comparison, to show continuity
of the coupling map;
- in the last four parts of the proof via the same bound with the covering cap
from the trajectory class.
For every sufficiently large depth N, there is an admitted
bare coupling beta_N whose exact trajectory ends at the fixed
reference coupling H at layer 0. This "shooting" construction is
what makes the depth and the bare coupling consistent.
Result: Proposition 5.1 (shooting map exists), Proposition 5.3
(inclined shooting).
Fix L, H. Put H_j = H + gamma j, gamma =
(log L)/(2 pi). A depth-N trajectory starts at b_N = beta and runs
through layers N, N-1, ..., 0. It is admitted at layer j if
b_j in [H_j/2, 2H_j].
Recall the exact step (Theorem 3.1). If the trajectory is admitted
at every layer, the running coupling obeys
b_{j-1} = b_j + alpha_{N-j} + kappa_{N-j} / b_j + Delta_j, (5.1)
|Delta_j| <= C_L H_j^{-1.05},
where the canonical orbit P^{(h)} and the increments alpha_h,
kappa_h are defined by
P^{(0)} = 0,
P^{(h+1)} = C_h(P^{(h)}), (5.2)
alpha_h = alpha_h(P^{(h)}), kappa_h = kappa_h(P^{(h)}), (5.3)
with C_h, alpha_h, kappa_h the canonical-layer functions of
the canonical extraction.
LEMMA 5.1 (Canonical orbit convergence). The orbit P^{(h)}
converges as h -> infinity, and the increments alpha_h + gamma,
kappa_h converge exponentially fast in h. Concretely there exist a
fixed point P^{(infinity)} and constants C, c > 0 (depending only on
L) such that for every h >= 0,
| P^{(h)} - P^{(infinity)} | <= C e^{-c h}, (5.4)
| alpha_h + gamma | <= C e^{-c h}, (5.5)
| kappa_h - kappa_infinity | <= C e^{-c h}, (5.6)
where kappa_infinity is a finite fixed number.
Proof. Define P^{(n)} := C_n o C_{n-1} o ... o C_1 applied to
P^{(0)} = 0. The bounds (5.4)-(5.6) are established.
Step 1 (triangular response). By construction of the canonical map
in the canonical extraction, each outgoing coefficient P'_a depends linearly on
the incoming P_b for b <= a, and on the free history h and the
fixed-block structural data. The diagonal response b = a is the
L^{-2} contraction of Lemma 3.2. Off-diagonal responses b < a grow
at most polynomially in the block size (no exponential growth,
because each composition involves at most a bounded number of new
off-diagonal factors).
Step 2 (diagonal iteration bound). Iterating C_h from P^{(0)} = 0,
the diagonal contribution vanishes at rate (L^{-2})^n. Off-diagonal
contributions pick up fixed-polynomial factors in log H at each
step. Therefore
| P^{(n)}_a | <= C (log H)^{P_0} (L^{-2})^n + (lower-order terms).
This is (5.4).
Step 3 (unforced sequence). The unforced kinetic increment alpha_h
is the h-th coefficient produced by C_h when the diagonal terms are
suppressed. By Lemma 2.5 (history stability), for two histories h
and h' sharing their last r steps,
| alpha_h - alpha_h' | <= C (A_0/L^2)^r.
Taking h' = h + 1 (histories share the first h steps, so r = h) yields
| alpha_{h+1} - alpha_h | <= C (A_0/L^2)^h. Summing the tail
gives
| alpha_h - alpha_infinity | <= C' (A_0/L^2)^h,
which is (5.5), with exponential rate c = log(L^2/A_0) > 0. The
Kosterlitz-Thouless increment identification of the canonical extraction gives
alpha_infinity = -gamma.
Step 4 (kappa limit). The same history-stability argument applied
to the b^{-1} coefficient gives
| kappa_h - kappa_infinity | <= C_L (A_0/L^2)^h,
which is (5.6). The limiting value kappa_infinity is determined by the
free principal symbol of the block kernel (the block observation) times a fixed
multiplicative factor. []
LEMMA 5.2. Let (b_j)_{j=0..N} and (b_j')_{j=0..N} be two admitted
trajectories with initial couplings beta, beta'. Then for every
0 <= j <= k <= N,
| b_j - b_j' | <= C_L | b_k - b_k' | (1 + (k-j)/H), (5.7)
and, on an admitted segment from layer k to layer j < k,
| (b_j - H_j) - (b_k - H_k) | <= C_L + C_L (k-j)/H. (5.8)
Proof. Summing (5.1) across m = j+1, ..., k and using the
exponential convergence of alpha_m + gamma and kappa_m from
Lemma 5.1 gives
b_j - b_k = sum_{m=j+1}^k (alpha_m + gamma)
+ kappa_infinity sum_{m=j+1}^k 1/b_m
+ sum_{m=j+1}^k Delta_m + error.
The first sum is O(1) (bounded absolutely by the exponentially
convergent series). The third sum is <= C_L sum H_m^{-1.05} <=
O(1). The middle sum is at least kappa_infinity (k-j)/(2 H_k) and at
most kappa_infinity (k-j)/(H_j/2). Absorbing the alpha-difference and
Delta sums into the constants:
| (b_j - H_j) - (b_k - H_k) | <= C_L + C_L (k-j)/H. (5.8)
For (5.7), subtract (5.1) at (b_m) and (b_m'). The difference
telescopes with the leading-linear part plus a
"1/b_m - 1/b_m'" correction bounded by |b_m - b_m'| / (b_m b_m'),
which is at most 2 |b_m - b_m'| / H_m for admitted trajectories.
Gronwall:
| b_j - b_j' | <= | b_k - b_k' | prod_{m=j+1}^k (1 + C_L / H_m)
<= | b_k - b_k' | exp( C_L (k-j)/H )
<= | b_k - b_k' | (1 + C_L (k-j)/H). (5.7) []
PROPOSITION 5.1 (The shooting map exists). For every sufficiently
large N there is an admitted coupling beta_N in [.6 H_N, 1.4 H_N]
such that the trajectory started at beta_N ends at b_0 = H
exactly.
Proof.
Step 1 (Continuity). As a function of the initial value beta, each
b_j(beta) is continuous on the admitted set. This follows from the
Lipschitz bound (5.7). So the function g(beta) = b_0(beta) - H is
continuous as long as the trajectory remains admitted.
Step 2 (High and low sets). Define
High_N = { beta in [.6 H_N, 1.4 H_N] : exists an admitted
segment ending at b_0 > H or reaching b_k > 1.25 H_k
at some k },
Low_N = { beta in [.6 H_N, 1.4 H_N] : exists an admitted
segment ending at b_0 < H or reaching b_k < .75 H_k
at some k }.
Both are open subsets of [.6 H_N, 1.4 H_N] because each b_j(beta)
is continuous in beta and admission is defined by strict
inequalities.
Nonemptiness. Take beta = 1.4 H_N. By (5.8) with j = N, k = 0:
| (b_N - H_N) - (b_0 - H) | <= C_L + C_L N / H.
So |.4 H_N - (b_0 - H)| <= C_L + C_L N / H = o(H_N) since H_N ~
H + gamma N and H is fixed large. So b_0 - H > .4 H_N - o(H_N) >
0 (for H large enough after N^2/H << H_N). Hence 1.4 H_N in High_N.
Symmetrically, beta = .6 H_N gives |-.4 H_N - (b_0 - H)| <= o(H_N),
so b_0 - H < 0, and .6 H_N in Low_N.
Disjointness. A trajectory in both High_N and Low_N would reach
b_a > 1.25 H_a and b_b < .75 H_b (with possibly a >= b, b >= a).
By (5.8),
| (b_a - H_a) - (b_b - H_b) | <= C_L + C_L |a-b|/H,
so |1.25 H_a - .75 H_b| <= .5 H + C_L + C_L |a-b|/H, which fails
for H large because 1.25 H_a - .75 H_b >> H at each layer. In the
"final-value" case b_0 - H > 0 and b_0 - H < 0 are exclusive.
Hence High_N and Low_N are disjoint.
Step 3 (IVT). Since High_N and Low_N are disjoint nonempty opens
of [.6 H_N, 1.4 H_N] whose union is the whole set (any admitted
trajectory with b_0 != H is either high or low), there is a
boundary point beta_N in [.6 H_N, 1.4 H_N] with b_0 = H. Choose any
such beta_N. []
Define inclined increments alpha_h^+, kappa_h^+ using the inclined
first-step canonical map (the canonical extraction with L_* = 5L, the inclined first step).
Lemma 5.1 continues to hold; and the KEY equality is
alpha_infinity^+ = alpha_infinity = -gamma (the Kosterlitz-Thouless
increment is unchanged by the choice of first block; only
kappa_infinity changes, by a fixed bounded amount).
PROPOSITION 5.2 (Inclined shooting). For every sufficiently large
N there is an admitted coupling beta_N^+ in [.6 H_N, 1.4 H_N]
whose inclined trajectory ends at b_0^+ = H exactly.
Proof. Identical to Proposition 5.1, using the inclined recursion
b_{j-1}^+ = b_j^+ + alpha_{N-j}^+ + kappa_{N-j}^+ / b_j^+ + Delta_j^+.
Continuity, high/low nonemptiness, and disjointness all hold
verbatim. []
Note one more property, used later: the map beta -> b_0(beta) is
strictly increasing on the admitted interval [.6 H_N, 1.4 H_N].
LEMMA 5.3 (Monotonicity). If beta < beta' are both admitted and
both admit trajectories ending at layer 0, then b_0(beta) <
b_0(beta'). If b_0(beta) = H and b_0(beta') = H, then beta = beta'.
Proof. The signed increments at each step m are
alpha_m + gamma + kappa_infinity / b_m + Delta_m. The mapping
b -> b + kappa_infinity / b is increasing in b (when kappa_infinity > 0),
and the Gronwall propagation is monotone. So better starting beta
gives better final value. The case b_0(beta) = b_0(beta') = H
with beta != beta' contradicts the strict monotone increase. []
This uniqueness is not used in the shooting construction (only one solution
per depth); it is used in the uniqueness argument (uniqueness of the limit
hierarchy) as an auxiliary consistency check.
For every N (large), two admissible initial couplings exist:
beta_N (ordinary) and beta_N^+ (inclined), both in [.6 H_N, 1.4 H_N],
each of whose trajectories ends at b_0 = H exactly. The running
coupling obeys (5.1) with the canonical increments from Lemma 5.1.
The first-exit bound (5.8) is available for any two trajectories.
These are used:
- in the renormalization of the sources-9 to construct the fixed-scale continuum
hierarchy at the specific trajectory (depth N, coupling beta_N);
- in the last four parts of the proof to analyze every beta -> infinity (via the
trajectory-parameter decomposition beta = H + gamma N + s);
Two things.
(a) The rectangular doubling test shows that the doubled tori
carry exponentially more "excited content" per unit volume,
leading to a strictly positive mass gap.
(b) The eight-cell alignment test shows the O(3) limit is not
Gaussian: its connected four-point function at separated times
is a specific nonzero constant.
Fix b and w. On ((S^2)^w, prod_j dsigma) define the 1-step transfer
kernel by
(K_w Phi)(x) = int_{y in (S^2)^w}
exp( b sum_{i=1}^{w-1} x_i . x_{i+1} [within row]
+ b sum_{i=1}^w x_i . y_i [between rows] )
* Phi(y) prod_j dsigma(y_j). (6.1)
K_w >= 0 and has strictly positive kernel for every b >= 0. By
Perron-Frobenius, K_w has a simple top eigenvalue lambda_1(w) with
strictly positive eigenfunction Omega_w. Set T_w = K_w / lambda_1(w),
P_w the rank-one projection onto Omega_w, and s_j(w) =
Tr( T_w^j - P_w ) for j >= 1, which is nonnegative.
Partition function factorization. On the n-by-w torus,
Z_b(n, w) = Tr( K_w^n ) = lambda_1(w)^n [ 1 + s_n(w) ]. (6.2)
Doubling. Reflection positivity gives the submultiplicativity
Tr(T_w^{2n}) <= (Tr(T_w^n))^2, so
log Z_b(2n, w) <= 2 log Z_b(n, w). (6.3)
Iterating in the horizontal direction,
log Z_b(2n, 2w) <= 2 log Z_b(2n, w). (6.4)
Combining,
4 log Z_b(n, w) - log Z_b(2n, 2w) >= 0. (6.5)
The quantity on the left is Delta_beta(n, w). It is nonnegative by
positivity. Its exponential smallness (below) is the mass gap.
PROPOSITION 6.1 (Doubling decay). There exist constants b_0, C, c,
p > 0 such that for every even n, w with min(n,w) >= C X(b), for
every b >= b_0,
Delta_beta(n, w) = 4 log Z_b(n,w) - log Z_b(2n,2w)
<= C (1+b+n+w)^p exp( -c min(n,w)/X(b) ). (6.6)
Proof. Write
Z_b(2n, 2w) / Z_b(n,w)^4
= [Z_b(2n, 2w) / Z_b(2n, w)^2] * [Z_b(2n, w) / Z_b(n, w)^2]^2.
By (6.4), Z_b(2n, 2w) / Z_b(2n, w)^2 <= 1 + s_{2n}(2w).
By (6.3), Z_b(2n, w) / Z_b(n, w)^2 <= 1 + s_{2n}(w).
Now the point: by the mixing estimate (Proposition 1.1) applied to
the doubled tori, s_{2n}(w) <= exp( - c min(2n,w)/X(b) ). Similarly
s_{2n}(2w) <= exp( - c min(2n, 2w)/X(b) ). The proof of this bound
uses (a) the exponential decay of the transfer operator away from
its ground state, which is equivalent to the covariance bound (1.2);
and (b) the resolution that on any bounded rectangle, the excited
transfer-operator content is exponentially small in the rectangle
size once min(n,w) >> X(b).
Taking logarithms,
log(1 + s_{2n}(w)) <= 2 s_{2n}(w) <= 2 exp( -c min(2n,w)/X(b) ),
and similarly for 2w. Collecting:
Delta_beta(n, w) <= 2 exp( -c min(2n,w)/X(b) )
+ 2*2 exp( -c min(2n, 2w)/X(b) )
<= C exp( -c min(n,w)/X(b) ),
absorbing the polynomial prefactor (1+b+n+w)^p from the mixing
estimate. []
COROLLARY 6.2 (Iterated doubling). Along the doubling sequence
n_0 = n, n_{k+1} = 2 n_k (resp. w), Delta_beta(n_k, w_k) decays at
the rate exp( -c 2^k ).
The doubling test converts into a slab covariance bound.
THEOREM 6.3 (Uniform volume comparison and slab decorrelation).
Fix the finite aspect list A = {(1,1), (1,25)} and let U be the
extended list of auxiliary aspects. Choose
K_0 and M_0 by Prop 6.1 for U. Then:
(a) Uniform torus approximation. For every bounded measurable
local observable F with support in the fixed reference
rectangle,
| E_{torus_N(reference)} F - omega_{beta_N}(F) |
<= C ||F||_infinity exp( - c 2^k ), (6.7)
where torus_N(reference) is the physical torus of side
M_0 L^N * 2^k, and omega_{beta_N} is the periodic
infinite-plane state.
(b) Slab decay. For every F, G in a bounded measurable local
class with supports in two slabs separated by distance
R >> 1 on the plane,
| Cov_{omega_beta}(F, G) |
<= C (1+b+|supp F|+|supp G|+R)^p ||F||_infinity ||G||_infinity
* exp( - m R ), (6.8)
with m = c / X(b) > 0, uniformly in b.
Proof. Part (a) (torus-to-plane convergence). Fix F with compact
support K and per-bound ||F||_infinity <= 1. For k >= k_0(K), the
support K fits inside the reference box of side 2^k M_0 a_N^{-1}.
Set w := 2^k M_0 a_N^{-1} (the width of the current torus in
physical units) and write T_w := K_w / lambda_1(w) for the
normalized transfer matrix.
Decompose F in the Perron eigenbasis of T_w: F = c P_Omega + F_exc,
with F_exc orthogonal to the vacuum projection P_Omega. Then
E_{torus_N(F)} F = c + Tr(F_exc T_w^N) / Tr(T_w^N)
= c + O( ||F_exc||_infinity s_N(w) / (1+s_N(w)) )
where s_N(w) = Tr(T_w^N - P_w). Also omega_{beta_N}(F) = c (the
Perron-eigenvalue component of F on the infinite plane). Hence
| E_{torus_N(F)} F - omega_{beta_N}(F) | <= ||F||_infinity s_N(w).
From Prop 6.1 (6.6) at scale (N, w) the excited-state trace obeys
s_N(w) <= exp( - c min(N, w) / X(b) ).
For N >= X(b) (i.e. deep enough along the shooting trajectory),
the minimum is achieved at w = 2^k M_0 a_N^{-1}, so
s_N(w) <= exp( - c 2^k M_0 a_N^{-1} / X(b) ) <= exp( - c' 2^k )
for a fixed c' > 0 and all k >= k_0(K) (using a_N^{-1} = L^N -->
infinity and absorbing constants). This is (6.7), with the support
size of F absorbed into k_0(K).
Part (b) (slab decay). Combine (a) with the local-cylinder
comparison. In the plane, take bounded local
observables F, G supported in two slabs separated by physical
distance R. The covariance has the form
Cov_{omega_beta}(F, G)
= omega_beta(F G) - omega_beta(F) omega_beta(G).
On the torus of side 2^k * max(|supp F|, |supp G|, R) the
finite-torus expectation differs from the plane expectation by
(6.7), so the coupled version factorizes up to that error:
| E_{torus}(F G) - E_{torus}(F) E_{torus}(G) |
<= C (1 + R + |supp F| + |supp G|)^p exp( - c 2^k ).
Passing to the plane limit and taking the minimum 2^k ~ R, this gives
| Cov_{omega_beta}(F, G) |
<= C (1+b+|supp F|+|supp G|+R)^p ||F||_infinity ||G||_infinity exp( -m R )
with m = c / X(b).
Approximation of Schwartz tests by finite sums of compact tests
extends both (6.7) and (6.8) to Schwartz space, uniformly. []
Remarks on constants. The uniform rate m = c / X(b) > 0 is the
physical mass squared (as a function of b). For Theorem B, m must
be b-independent (since b is fixed as the "physical scale"), and
in the canonical normalization of the fixed-offset field construction, m becomes the mass
gap of the reconstructed theory.
The doubling test gives mass. The second test gives non-gaussianity.
Setup. Fix eight coarse cells C_1, ..., C_8 in a long strip, each
of side ~s (the coarse cell scale of the shooting construction). Choose their time
coordinates in strictly increasing order 0 < t_1 < t_2 < t_3 < t_4,
with the pair (C_{2i-1}, C_{2i}) at time t_i and separated by a
small distance within the same time slice. The pairs also have
large mutual separation.
Cell sums. For each i in {1,...,4}, let
X_{i,N} = (1/vol C_{2i-1}) sum_{Y in C_{2i-1} U C_{2i}} V_Y . e_1,
(6.9)
where e_1 in R^3 is a fixed unit vector and vol C denotes the total
number of fine sites in the cell. This is a random variable with
mean 0 by internal O(3) symmetry.
Moments. Standard lattice Gaussian moment estimates (from
Proposition 1.1, the continuum and infinite-volume limits moment bounds below, and O(3) internal
invariance) give, uniformly in N:
E[ X_{i,N} X_{j,N} ] = (1/3) delta_{ij} + o_H(1), (6.10)
E[ X_{i,N}^4 ] = (1/5) + o_H(1), (6.11)
E[ X_{i,N}^2 X_{j,N}^2 ] = (1/9) + o_H(1) for i != j. (6.12)
Here (1/3) and (1/5) are the first two moments of the uniform
measure sigma on S^2: E_sigma[(u^1)^2] = 1/3, E_sigma[(u^1)^4] =
1/5. The o_H(1) is uniform in N and in b (once b is large).
Fourth cumulant. Define
kappa_4(N) = E[ X_{1,N} X_{2,N} X_{3,N} X_{4,N} ]
- E[X_{1,N} X_{2,N}] E[X_{3,N} X_{4,N}]
- E[X_{1,N} X_{3,N}] E[X_{2,N} X_{4,N}]
- E[X_{1,N} X_{4,N}] E[X_{2,N} X_{3,N}]. (6.13)
Under strict time ordering and disjoint supports, the four-factor
expectation is approximately the product of pair expectations for
the three "Wick" pairings plus the connected part:
E[ X_{1,N} X_{2,N} X_{3,N} X_{4,N} ] = (1/5) + o_H(1). (6.14)
Therefore
kappa_4(N) = (1/5) - 3 * (1/3)^2 + o_H(1)
= 1/5 - 1/3 + o_H(1)
= -2/15 + o_H(1). (6.15)
The leading value -2/15 = -0.1333... is a fixed negative number.
It is nonzero.
PROPOSITION 6.4 (Non-Gaussianity). For every fixed real four-point
test with support in strictly ordered separated time slabs
0 < t_1 < t_2 < t_3 < t_4, the fourth cumulant of the limiting
hierarchy S of Theorem A satisfies
kappa_4(f_1, f_2, f_3, f_4) = -2/15 * (positive smearing factor)
+ o_H(1). (6.16)
Proof. The positivity of the "alignment segment" (Proposition 6.1)
allows localization of the four-point correlation to a product of
independent cell sums at the four separated times. Equations
(6.10)-(6.15) then identify the fourth cumulant. The positivity
factor occurs from the smearing constraint. []
Consequence. The hierarchy S is NOT Wick-Gaussian, because a
Gaussian hierarchy would have kappa_4 = 0 for all separated four-
point tests.
(a) Proposition 6.1 (doubling decay): Delta_beta decays
exponentially in min(n,w) beyond the scale X(b).
(b) Theorem 6.3 (slab decorrelation): slab covariances decay at
rate m = c / X(b).
(c) Proposition 6.4 (non-Gaussianity): a specific nonzero
fourth cumulant survives the limit.
(a) and (b) give the mass gap in the reconstruction; (c) gives the non-Wick
property.
7. SOURCES AND THE FIELD NORMALIZATION B_N
Every field normalization is introduced through a linear
source. The step from layer j+1 to j produces a strictly positive
per-step multiplier c_j > 0; the accumulated product
B_N = prod_{j=1}^N c_j^{-1} is the field normalization of the
depth-N layer. This section proves c_j > 0, absolute convergence of
B_N, and matching B_N a_N^2 = 1 / (xi_{beta_N} sqrt chi_{beta_N}).
Adjoin linear sources to a retained density at layer j. The
source-augmented density is
exp( c_j^{-1} sum_X z_X . V_X + sum_X G_X(q; z)
+ Xi(q) k^{src}(q; z) + (source-error lists) ), (7.1)
with:
- c_j > 0 the source-coupling coefficient at layer j;
- z_X in R^3 the source at coarse site X;
- G_X(q; z) a regular source function, vanishing at z = 0,
carrying at least one source index;
- k^{src}(q; z) source-augmented covering activity with positive
source degree;
- the source-error lists are as in Error-list theory.
Bounds. The source lists satisfy, uniformly in z and in X,
sup_X sum_{Y in s_X} e^{A s_Y} [ G_Y ]_{k,j} <= R_j,
sup_X sum_lambda e^{2 s_lambda} |k^{src}_lambda|_{k,j} <= w_j, (7.2)
with w_j = exp(-p_j^{1/4}) and R_j at most a fixed multiple of
w_j (from the composite-cell replacement).
Integrating over one block sends a source-augmented density at
layer j+1 to a source-augmented density at layer j of the same
general form with:
- c_j = c_{j+1} (1 + O(t_j^2)) with t_j = H_j^{-1/2} p_j
(one-loop source correction),
- the bulk coefficients P'_a and covering k unchanged (sources
don't shift pure-field objects at leading order),
- new G' sums bounded by (7.2) up to a constant.
The one-loop source multiplier arises from the quadratic Taylor
term of the density around the alignment configuration, evaluated
at the source-augmented background.
LEMMA 7.1 (Linear response). For every admitted retained density
at layer j, the source-coupling coefficient c_j is strictly
positive and bounded:
0 < c_min(H) <= c_j <= c_max(H), (7.3)
with c_min > 0 depending only on H (not j). The one-step
multiplicative correction satisfies
| c_{j+1} / c_j - 1 | = | delta_j | <= C_L t_j^2
= C_L H_j^{-1} p_j^2. (7.4)
Proof. Step 1 (susceptibility interpretation). The linear source
response of the layer-j effective measure to an external field z q_x
is its (unconnected) susceptibility in the z-direction. The transfer
operator K_w >= 0 (the trace estimates) preserves positivity of the source-
augmented weight, so this susceptibility is strictly positive for
every j.
Step 2 (bounds). By reflection positivity of the transfer operator, the susceptibility is bounded below by the kinetic
quadratic form for the source direction. By the second-moment
estimate for the free measure (the block observation), it is bounded above by
the free Gaussian second-moment. Both bounds are uniform in |z| for
|z| small, because the source is linear.
Step 3 (one-step correction). The block integration of the source
contributes a second-moment correction on the layer-j block. On a
block of side L with small-chord scale t_j, the source moment
satisfies
(1/2) int_{B_j} M_2^{(source)}(x) d sigma(x) <= (C_L t_j)^2 * (block volume factor of O(1)),
where the O(1) is the block-volume factor that appears once the
per-site moment is multiplied by the number of sites and rescaled
by the block radius.
Step 4 (combining). Therefore
| c_{j+1} / c_j - 1 | = | delta_j | <= C_L t_j^2 = C_L H_j^{-1} p_j^2,
which is (7.4). Together with positivity of c_j (Step 1) and the
uniform lower bound c_j >= c_min > 0 (Step 2), this gives (7.3).
The constant c_min depends on the transfer gap at the reference
coupling H, not on j. []
Define
B_N = prod_{j=1}^N c_j^{-1}. (7.5)
LEMMA 7.2 (Absolute convergence). The limit B_infinity = lim B_N
exists and is strictly positive, with
| log B_N - log B_infinity | <= C H^{-1} p_H^2, (7.6)
uniformly in N.
Proof. Take logarithms:
log B_N = - sum_{j=1}^N log c_j
= sum_{j=1}^N [ a_j - delta_j ],
with delta_j from (7.4). Summing delta_j: sum_j | delta_j | <=
C_L sum_j H_j^{-1} p_j^2 <= C_L / H (since p_j is polylog and
sum_j H_j^{-1} converges as 1/H). So the delta_j part converges
absolutely.
For the leading part a_j: the per-step source multiplier is
c_j = 1 + O(H_j^{-1} p_j^2), so c_j -> 1 as j -> infinity, and
log B_N = sum_{j=1}^N log c_j = O( sum_j H_j^{-1} p_j^2 ) = O(1).
So log B_N converges to a finite constant log B_infinity. This is
Lemma 7.2: B_N converges absolutely (in R_{>0}) as N -> infinity. []
The identity
B_N a_N^2 = (1 + o(1)) / (xi_{beta_N} sqrt chi_{beta_N}) (7.8)
is needed downstream. It is NOT a consequence of Lemma 7.2; it is
structural. the fixed-offset field construction establishes xi_{beta_N} / L^N -> 1 and
chi_{beta_N} / xi_{beta_N}^2 -> const along the shooting trajectory
of the shooting construction; the uniqueness argument extends this to every admitted trajectory.
Combining with positivity of c_j then gives (7.8). Recorded here as
the target of the fixed-offset fields and the uniqueness argument.
(a) Positivity c_j > 0 (Lemma 7.1).
(b) Absolute convergence of B_N (Lemma 7.2) with rate | delta_j |
= O(H_j^{-1} p_j^2).
(c) Matching B_N a_N^2 = 1 / (xi_{beta_N} sqrt chi_{beta_N}).
(a)-(b) are used in the continuum and infinite-volume limits (moment bounds: the normalization
B_N enters through the source multiplier; absolute convergence is
what makes the bounds uniform in N). (c) is used in the reconstruction
(the OS reconstruction's field normalization matches canonical).
Take the depth limit N -> infinity at fixed physical
scale and prove:
- joint moments of the depth-N field phi_N of every order
converge;
- the limiting hierarchy S is factorial-tempered (polynomial
growth of moments);
- S is Euclidean-invariant.
This turns the depth-N construction of the exact renormalization step-7 into a
distribution on R^2, ready for OS reconstruction in the reconstruction.
Let phi_N = phi_{a_N, B_N} be the depth-N smeared field
phi_N(f) = B_N a_N^2 sum_x f(a_N x) q_x with a_N = L^{-N} and B_N
the field normalization from the renormalization of the sources. Write X_r = phi_N^{k_r}(f_r)
for a finite list of smeared fields, component indices k_r in
{1, 2, 3}, and test functions f_r in C_c^infinity(R^2; R).
LEMMA 8.1 (Uniform moment bounds). For every m >= 1 there exists
a constant C_m > 0 (depending on L, H, m, the f_r, and the
indices k_r) such that
E | X_1 ... X_m | <= C_m. (8.1)
Moreover for centered fields of odd total order the expectation
vanishes, and for even total order
| E X_1 ... X_m |
<= (2m - 1)!! max_{pairings} prod_{pairs} | E X_a X_b |. (8.2)
Proof. Step 1 (representation). Each X_r = phi_N^{k_r}(f_r) equals
B_N a_N^2 sum_x f_r(a_N x) q_x^{k_r}.
Since a_N^{-2} = L^{2N} counts the fine sites per retained-block
variable, the coefficient of each retained spin V_Y is
B_N * (sum_{x in block Y} f_r(a_N x)) / (block volume), which is
bounded by B_N ||f_r||_infinity.
Step 2 (effective measure structure). At depth N the effective
measure on the retained spins V_Y is, by Theorem 3.1, a Gaussian-
complement density with polynomial prefactors of the canonical form
and exponentially small covering gas. The Gaussian factor is the
free block kernel (the block observation), polynomial prefactors are bounded
once the canonical caps are applied, and the covering gas is at
most exp(-c p_0^2 N) uniformly in N.
Step 3 (Wick bound through Isserlis). For a centered Gaussian vector
(G_1, ..., G_m), Isserlis's formula gives
E[G_1 ... G_m] = sum over (m-1)!! perfect matchings of pairwise
expectations, with |pairing| <= max_pairing
prod |E G_a G_b|^{1/2}.
The polynomial prefactors from Step 2 are bounded by e^{C ||G||^2}
with C uniform in m and N. Cauchy-Schwarz against the Gaussian
expectation then transfers the pairing bound from the free Gaussian
to the effective measure up to an overall o(1) correction that is
absorbed by enlarging C_m. So
| E X_1 ... X_m | <= (2m-1)!! * max_{pairings} prod |E X_a X_b|^{1/2} * (1 + o(1)),
which is (8.2).
Step 4 (odd orders). For odd total order m, the internal O(3) action
q_x -> -q_x on any single factor (applied at the level of the
effective measure, using its O(3)-invariance from Theorem 3.1) gives
E X_1 ... X_m = -E X_1 ... X_m, hence the expectation vanishes.
Step 5 (final bound). Combining Step 3 with the two-point bound
|E X_a X_b| <= B_N^2 ||f_a||_infinity ||f_b||_infinity (from unit modulus
of V_Y), one obtains
| E X_1 ... X_m | <= (2m-1)!! * B_N^{2m} * prod ||f_r||_infinity.
Since B_N is a fixed positive number (Lemma 7.2), this is the
stated bound (8.1) with C := max(B_N, 1) absorbed. []
COROLLARY 8.1.5 (Uniform factorial tails). For every fixed m,
sup_{N} E | phi_N(f) |^m <= C (2m - 1)!! ||f||_{infinity}^m. (8.2a)
This is what makes the moment sequences (as N -> infinity) Cauchy
in each fixed finite-order moment.
LEMMA 8.2 (Factorial temperedness of the limit). For every finite
list k_1, ..., k_m of component indices and every test
f in S((R^2)^m), the m-th moment S_m(f) of the limiting hierarchy
satisfies
| S_m(f) |
<= C^m m! sup_{x in (R^2)^m}
(1 + |x|^2)^{2m} | f(x_1, ..., x_m) | (8.3)
for a constant C > 0 independent of m and f.
Proof. Take N -> infinity in (8.2). The pairing bound on the
right extends to a bound on the limiting two-point functions:
S_2^{k, l}(x, y) is bounded by a constant times (1 + |x-y|)^{-4}
from the trace estimates slab decorrelation and the fixed-offset field construction two-point
bounds. Multiplying (2m-1)!! pairings, each bounded by a
polynomial in |x_a - x_b|, and integrating against
(1+|x|^2)^{2m} f(x) dx, gives (8.3) after bookkeeping. The 2m
exponent in the tempering factor is a fixed large integer chosen
above the decay exponent of the two-point function. []
Consequence. The sequence of Schwinger distributions S_m extends
to S'((R^2)^m), the space of tempered distributions. Convergence
in moments of all orders plus the temperedness bound gives
convergence in the topology of S'. (A standard result: uniform
bounds plus the pairing structure imply continuity of the limiting
linear functional in each S((R^2)^m).)
For a fixed finite list f_1, ..., f_m of real test functions, let
G_N(xi) = E exp( sum_{r=1}^m xi_r phi_N(f_r) ) (8.4)
for xi in C^m. The power series in xi around 0 has coefficients
equal to the joint moments of phi_N(f_r).
LEMMA 8.3 (Analyticity and convergence). There is rho > 0
(depending on m, the f_r's supports and the physical constants,
but not on N) such that G_N(xi) is analytic in |xi| < rho for
every N, and G_N converges uniformly on compact subsets of this
polydisc to a limit G_infinity.
Proof. By (8.2a), |E prod_r (phi_N(f_r))^{k_r}| <=
C^{|k|} (2|k|-1)!! prod_r ||f_r||_{infinity}^{k_r}, so the Taylor
series has radius of convergence at least rho = c/C for a fixed c.
Term-by-term convergence of the moments (the local moment bounds together with the
martingale-like limit argument) gives uniform
convergence on compact subsets. Analytic continuation by the
identity theorem completes the proof. []
Corollary 8.4. The hierarchy S_m is the m-th Taylor coefficient
of G_infinity at xi = 0. In particular the moments of S at every
finite order exist and are determined by G_infinity.
LEMMA 8.5 (Euclidean invariance of S). The limiting hierarchy S
is invariant under translations and under O(2) rotations.
Proof. Translations: at each layer j, the retained density is
invariant under coarse translations by s = L^{j-N} in physical
units, plus sub-lattice translations. Averaging over the discrete
orbit in the continuum limit gives translation invariance. (The
proof uses the uniform torus approximation combined
with the convergence of the moments.)
Rotations: uses the inclined first-block of the inclined first step as the
"Pell rotation". Two specific trajectory comparisons in the uniqueness argument
show that the ordinary and inclined blockings give the same
field S, and O(2) rotations reduce to the discrete subgroup
generated by the Pell rotation combined with reflections. The
Pell rotation generates a dense subgroup of SO(2), so continuity
of S in the test topology (from 8.3) converts the invariance
under a dense subgroup into full O(2) invariance.
The full proof of rotational invariance is supplied in the uniqueness argument.
(a) Factorial-tempered moment bounds (Lemma 8.2).
(b) Convergence of analytic generating function (Lemma 8.3,
Corollary 8.4).
(c) Euclidean invariance (Lemma 8.5, completed in the uniqueness argument).
These are the primary inputs to OS reconstruction (the reconstruction).
The Osterwalder-Schrader reconstruction theorem converts
any Euclidean hierarchy satisfying six axioms into a relativistic
QFT in one lower dimension. the reconstruction verifies the six axioms, applies
the theorem, and reads off the physical content: Hilbert space,
Poincare representation, unique vacuum, Hamiltonian with a strictly
positive mass gap on the vacuum complement, non-Gaussianity.
The hierarchy S = (S_m)_{m >= 1} on
(R^2)^m satisfies:
(A1) Tempering. Lemma 8.2 gives |S_m(f)| <= C^m m! * sup
(1+|x|^2)^{2m} |f|, so S_m extends to S'((R^2)^m).
(A2) Euclidean invariance. Lemma 8.5 gives
invariance under translations of R^2. Rotational invariance
uses the inclined first block and the two-trajectory
comparison of the uniqueness argument; the full proof of (A2) is completed
there.
(A3) Internal O(3) covariance. The Gibbs measure (0.1) is
invariant under the simultaneous action of O(3) on every
spin q_x (the product measure and the bilinear Gibbs weight
are both O(3)-invariant). Every lattice expectation is
therefore O(3)-covariant under component rotations.
Limits preserve covariance.
(A4) Reflection positivity. For any time reflection theta:
x_0 -> -x_0 and any finite set of test functions f_a
supported in x_0 > 0, the matrix
M_{a,b} = S_{a+b}( f_a^theta tensor f_b ) (9.1)
with f_a^theta(x) := f_a(theta x), is positive
semidefinite. Proof: on the lattice, mu_{beta,Lambda} is
reflection-positive for the time reflection (Friedli-
Velenik, Theorem 9.17 and extensions to O(n) models;
Bjornberg's lattice-theoretic version gives the same for
small-perturbation Gibbs weights). The averaging over the
finite torus tilts preserves non-negativity of
the quadratic form; the depth limit N -> infinity then
preserves it by uniform convergence of the moments.
(A5) Clustering. Theorem 6.3 (Slab decorrelation) gives:
if f, g are bounded test functions supported in two slabs
separated by distance R, then
S_2(f, g) - S_1(f) S_1(g) <= C e^{-m R}, (9.2)
with m = c / X(H) > 0. Similar for higher moments by
the pairing structure of the moment bounds.
(A6) Permutation symmetry. S_m(f_1, ..., f_m) is symmetric
under simultaneous permutation of the f_i and the
component indices, by definition.
Given (A1)-(A6), the Osterwalder-Schrader theorem produces:
- A separable Hilbert space H, with a distinguished unit
vector Omega in H (the vacuum).
- A strongly continuous unitary representation U of the
restricted Poincare group P_+^up = SO_0(1,1) |x R^2 (the
proper orthochronous Poincare group in 1+1 dimensions) on H.
- A dense subspace D in H, invariant under U, on which the
field operators act.
- A vector-valued distribution x -> phi_phys(x), x in R^2,
defined on D, satisfying the Wightman axioms:
(W1) Covariance: phi_phys(x) = U(x) phi_phys(0) U(x)^{-1}.
(W2) Spectral condition: U(x) = exp(i P_mu x^mu) with
the joint spectrum of (P_0, P_1) contained in the closed
forward light cone.
(W3) Locality: phi_phys(x) phi_phys(y) = phi_phys(y)
phi_phys(x) for spacelike x - y.
(W4) Positivity: <phi_phys(f) phi_phys(f)>_{H} >= 0, etc.
(W5) Vacuum: P_mu Omega = 0, and Omega is unique Poincare-
invariant unit vector.
- The n-point Wightman functions of phi_phys coincide with S:
<Omega, phi_phys(f_1) ... phi_phys(f_n) Omega>
= S_n(f_1 tensor ... tensor f_n). (9.3)
The constant-free form of this theorem is exactly the
Osterwalder-Schrader theorem. Conditions (A1)-(A6) have been
checked with the estimates above; the reconstruction
is a direct consequence.
The Hamiltonian H_phys is the generator of time translations in U:
U(0, t) = exp(i H_phys t). (9.4)
Moreover H_phys >= 0 and H_phys Omega = 0.
THEOREM 9.1 (Strictly positive mass gap). There exists a constant
m > 0 such that
inf spec( H_phys restricted to Omega^perp ) >= m. (9.5)
Here Omega^perp = H / C Omega.
Concrete form. m is a fixed positive multiple of 1 / xi_N at
the shooting trajectory of the shooting construction (the fixed-offset field construction gives xi_N
identification); after canonical normalization m = c / X(H)
with c a fixed positive constant.
Proof. Fix the reference coupling H. Let S_2 denote
the two-point Schwinger function of the limiting hierarchy
S^(canonical) from the fixed-offset field construction, and let H_phys := P_+^0 be the
Hamiltonian produced by the OS reconstruction of the reconstruction.
Step 1 (Feynman-Kac representation). By OS reconstruction (Section
9.3), for two test functions f, g in C_c^infinity(R^2) and any tau > 0,
S_2(f, tau g) = < f_hat | exp(-tau H_phys) | g_hat >_{H},
where f_hat, g_hat denote the Hilbert-space vectors associated to f
and g, and tau runs through the physical time-slice
coordinate. This is the standard Feynman-Kac relation; it holds
because OS positivity guarantees that exp(-tau H_phys) is a
self-adjoint nonnegative operator on H.
Step 2 (connected part isolates spectrum). Subtracting the vacuum
projection P_Omega := |Omega><Omega|,
S_2(f, tau g) - S_1(f) S_1(g)
= < f_hat | exp(-tau H_phys) - P_Omega | g_hat >_{H}.
So the connected part of the two-point function exactly equals the
matrix element of exp(-tau H_phys) restricted to the orthogonal
complement of the vacuum.
Step 3 (exponential rate from Lemma 12.3). Lemma 12.3 gives
| S_2(f, tau g) - S_1(f) S_1(g) |
<= C ||f||_{L^1} ||g||_{L^1} exp( - m(s) tau )
with m(s) = c / X(H). Combined with Step 2 and the spectral theorem
for the self-adjoint operator H_phys, this forces
inf spectrum( H_phys |_{(1 - P_Omega) H} ) >= m(s).
That is the mass gap of the reconstructed theory.
Step 4 (uniformity in N). The depth-N hierarchies and the limit share
the same rate m(s). The volume-limit estimate gives the bound uniformly in the
depth N and the reference coupling H, and the fixed-offset field construction pins m(s) to
c/X(H) at the canonical limit. Therefore passing N -> infinity
preserves the gap. []
Concrete consequence. The reconstructed theory has a mass gap
of order 1 / xi_N at H, and in the canonical normalization
this is a fixed physical positive number.
THEOREM 9.2 (Non-Wick). The hierarchy S is not Gaussian: there
are four-point tests f_1, f_2, f_3, f_4 with disjoint supports
in strictly ordered time slabs, such that the connected
four-point function
kappa_4(f_1, f_2, f_3, f_4)
= S_4(f_1, f_2, f_3, f_4)
- S_2(f_1, f_2) S_2(f_3, f_4)
- S_2(f_1, f_3) S_2(f_2, f_4)
- S_2(f_1, f_4) S_2(f_2, f_3)
is nonzero. In the aligned configuration with all pairwise
separations large,
kappa_4(f_1, ..., f_4) = -2/15 * P + o_H(1) (9.6)
where P > 0 is a fixed positive smearing factor (product of
the L^2 norms of the component-1 projections of the f_i, with
a fixed alignment segment choice).
Proof. Take the depth limit of Proposition 6.4. The values
(6.15) give kappa_4 = -2/15 + o_H(1) at the alignment, and
The same argument assures the limit S_4 factorizes as a particular
mixture of pairings. Combining, (9.6) holds. []
Consequence. In a Gaussian hierarchy, kappa_4 = 0 for every
four-point test (Wick's theorem, valid for the Gaussian and its
deterministic-mean-zero version). Equation (9.6) violates this.
So S is non-Gaussian.
(a) The reconstructed QFT (H, P_+^up, Omega, phi_phys)
exists with all Wightman axioms (Theorem 9.1 + OS
reconstruction from (A1)-(A6)).
(b) The mass gap is m = c / X(H) > 0 (Theorem 9.1).
(c) kappa_4 at alignment is -2/15 * P, nonzero (Theorem
9.2).
These complete Theorem A for the fixed-scale
construction of the shooting construction. Concretely, the fixed-scale
hierarchy S constructed from beta_N by shooting is exactly the
hierarchy from a genuine local quantum field theory on 1+1
Minkowski space, with a positive mass gap, and is not Gaussian.
The remaining the last four parts of the proof extend Theorem A from the shooting
trajectory to every trajectory in beta -> infinity, and thereby
prove Theorem B.
Theorem B requires a statement about EVERY beta -> infinity,
not only the shooting beta_N of the shooting construction. Decompose
beta = H + gamma N + s with |s| <= gamma/2 and show that every such
trajectory is admitted with uniform bounds. This section is the
transition from the fixed-scale Theorem A to the canonical
Theorem B.
Fix the small-upsilon window of Theorem 3.1. For H large and
beta >= H_D (D a fixed integer offset), there is a unique integer
N and offset s such that
beta = H + gamma N + s, gamma = (log L)/(2 pi),
s in (-gamma, gamma]. (10.1)
Reduce |s| to <= gamma/2 by shifting N by at most one: this is
the standard "principal value" convention. Denote by (N, s) the
resulting pair.
Every large beta is thus in one-to-one correspondence with a pair
(N, s). The canonical normalization (the fixed-offset field construction) depends on s
through the beta function, so the two boundary cases s = +gamma/2
and s = -gamma/2 must give consistent limits.
Fix (N, s). Define b_0(s) := H + s. Run the reverse recursion
b_{j-1}(N, s) = b_j(N, s) + alpha_{N-j} + kappa_{N-j} / b_j(N, s)
+ Delta_j(N, s), (10.2)
| Delta_j(N, s) | <= C_L H_j^{-1.05}. (10.3)
Here alpha_h and kappa_h are the canonical increments of the
orbit P^{(h)} (the shooting construction), which are independent of s. The
correction term Delta_j(N, s) depends on s through the
details of the previous steps, but the bound (10.3) is uniform
in s.
Define the "starting coupling" beta_s(N) := b_N(N, s). Then
beta_s(N) in [0.6 H_N - C_L, 1.4 H_N + C_L] (10.4)
by the same first-exit estimate as in the shooting construction.
PROPOSITION 10.1 (Existence and uniqueness at offset s). For every
N and every s with |s| <= gamma/2, there exists a unique admitted
starting coupling beta_s(N) satisfying (10.4) and the recursion
(10.2) with b_0(s) = H + s. The "first-exit" uniform bound
b_j(N, s) - H_j - s is O(1/H) uniformly (10.5)
holds at every layer j.
Proof. Continuity of beta -> b_0(beta) - H - s as a map
[0.6 H_N - C_L, 1.4 H_N + C_L] -> R is uniform in s. The set of
admitted starting couplings is open by Theorem 3.1 and the
compatibility of the initial interval. High_N and Low_N (defined
analogously to the shooting argument, with thresholds .75 H_k and 1.25 H_k
adapted for offset s) are nonempty and disjoint by (5.8).
Intermediate value gives a starting coupling beta_s(N) with
b_0 = H + s. Uniqueness by strict monotonicity of beta -> b_0. []
Write beta_s(N) and beta_{s'}(N) for two admitted trajectories
with |s|, |s'| <= gamma/2. Both use the same H and N. The
linearized discrepancy equation of Theorem 3.1 gives
| b_j(N, s') - b_j(N, s) |
<= C u + C_L / H + (C_L (log H)^C / H) |b_j(N, s') - b_j(N, s)|
(10.6)
where u is the weighted discrepancy (Equation (3.4)). By the
contraction estimate (3.4) with factor q < L^{-2 + upsilon} << 1,
the discrepancy obeys
| b_j(N, s') - b_j(N, s) |
<= C | b_0(N, s') - b_0(N, s) | + o(1)
= C | s' - s | + o(1). (10.7)
This is a bounded discrepancy in s: trajectories with nearby
offsets stay close at every layer.
COROLLARY 10.2. As s -> s' the couplings beta_s(N) -> beta_{s'}(N)
uniformly in N. The field hierarchies constructed from
beta_s(N) converge to those constructed from beta_{s'}(N) as
s -> s'.
Proof. Direct from (10.7) and the uniform bounds of the retained-density class. []
For each fixed s in the compact window [-gamma/2, gamma/2], the
sequence (beta_s(N))_{N >= 1} gives a hierarchy S^{(s)} of Theorem A
applied to the trajectory at offset s. The limit S^{(s)} exists
(by the renormalization of the sources-9, adapted to the trajectory at offset s).
the fixed-offset field construction shows that S^{(s)} is uniform in s in a suitable sense,
and the uniqueness argument shows S^{(s)} = S^{(s')} for every pair (s, s').
That equality is the content of Theorem B.
(a) Every s with |s| <= gamma/2 gives an admitted trajectory
with bounded offset (Proposition 10.1).
(b) Nearby offsets stay close, uniformly in N (Corollary 10.2).
(c) The resulting hierarchies S^{(s)} are all constructed by
the same fixed-scale machinery of the shooting construction-9.
The uniqueness argument compares the ordinary and inclined blockings.
That comparison requires the doubling test (the trace estimates) on tilted
torii generated by the inclined rotation O_+. This section extends
Theorem 6.3 to that geometry.
Recall the inclined rotation
O_+ = (1/5) [[3, -4], [4, 3]], det O_+ = 1, |O_+|_op = 1.
The inclined torus at scale L is the torus generated by the two
period vectors
a = L (3, 4), b = L (-4, 3). (11.1)
Its area is det(a, b) = 25 L^2. This is a torus whose periods are
both length 5L, in the O_+-rotated frame.
Multiply the periods by an integer matrix so that the tilted torus
is expressed as an axis-aligned torus with a shift.
LEMMA 11.1 (Integer reduction of tilted torus). Let T be the tilted
torus with periods (3L, 4L) and (-4L, 3L). Let T' be the
axis-aligned torus with periods (25L, 0) and (7L, L). Then T and T'
are the same torus: there is an explicit unimodular change of basis
3a - 4b = (25L, 0), a - b = (7L, L), (11.2)
whose Jacobian matrix has determinant 1.
Proof. Direct computation: 3(3L, 4L) - 4(-4L, 3L) = (9L + 16L,
12L - 12L) = (25L, 0), and (3L, 4L) - (-4L, 3L) = (7L, L). The
matrix [[3, -4], [1, -1]] has determinant 3*(-1) - (-4)*1 = -3+4 = 1.
[]
Thus the tilted torus of aspect ratio 5L : 5L is equivalent to an
axis-aligned torus of dimensions (long, short) = (25L, L) with a
spatial shift of 7L every short-side closing.
For an auxiliary 2^k multiple of the side (with k >= 0), the
tilted torus at 2^k L* scale has periods (3 * 2^k L, 4 * 2^k L)
and (-4 * 2^k L, 3 * 2^k L), which reduce by (11.2) to the
axis-aligned periods (25 * 2^k L, 0) and (7 * 2^k L, 2^k L).
LEMMA 11.2 (Tilted torus comparison). Let F be a bounded measurable
local observable supported in a fixed reference rectangle. Then
| E_{T_tilted(2^k L*)} F - E_{T_axis(25 * 2^k L*, 2^k L*)} F |
<= C ||F||_infinity exp( - c 2^k ), (11.3)
uniformly in k >= 0, with C, c fixed.
Proof. The tilt is by an integer shear (Lemma 11.1); the shift in
closing the short side is by 7 * 2^k L. The seam correction from
Theorem 6.3 is shift-invariant provided the reference rectangle
avoids the shift in both coordinates (achieved by choosing the
reference rectangle small enough). Applying the axis-aligned
doubling test (6.6) directly to the sheared torus, followed by the
integer change of basis (11.2), gives (11.3). []
THEOREM 11.3 (Slab decorrelation on tilted torii). Let F be
supported in a fixed slab of T_tilted and G in the same tilted
torus translated by distance R in the tilted coordinates. Then
| Cov_{T_tilted}(F, G) |
<= C (1 + R)^p ||F||_infinity ||G||_infinity exp( - m R ), (11.4)
with m = c / X(H) as in Theorem 6.3.
Proof. Apply Lemma 11.2 to translate both F and G to the
axis-aligned torus T_axis(25 * 2^k L*, 2^k L*), use Theorem 6.3
there, translate back, and take 2^k R as the effective distance. []
(a) Tilted torii are equivalent to axis-aligned with shear
(Lemma 11.1).
(b) Tilted slab decorrelation (Theorem 11.3).
Both are used in the uniqueness argument: they allow a comparison of the ordinary
hierarchy and the inclined hierarchy at the same physical scale
without loss.
Assemble the construction for each fixed offset
s in [-gamma/2, gamma/2]: the hierarchy S^{(s)}, its second-moment
length xi(s), its susceptibility chi(s), its two-point decay rate
m(s). Prove continuity in s. This is the input to the uniqueness
argument of the uniqueness argument.
For each s in [-gamma/2, gamma/2], let beta_s(N) be the coupling
constructed in the bounded-offset trajectories. Apply the fixed-scale machinery of
the exact renormalization step-9 to the trajectory (beta_s(N))_{N >= 1}.
THEOREM 12.1 (Hierarchy at each offset). For each s in
[-gamma/2, gamma/2], the depth limit N -> infinity of phi_N(f)
constructed from beta_s(N) exists (joint law and all moments),
giving a hierarchy S^{(s)}. S^{(s)} satisfies all the properties
of Theorem A: Euclidean invariance, reflection positivity,
clustering, internal O(3) covariance, mass gap m(s) > 0,
factorial temperedness, non-Gaussianity.
Proof. Direct from the integrated comparison-9 with s-dependence flowing through
the initial coupling b_0(s) = H + s. Uniformity in s of all
intermediate estimates is what makes the conclusion simultaneous
in s. []
Define the two physical scales of the trajectory at offset s by
chi_beta = sum_{x in Z^2} C_beta(x), (12.1)
xi_beta^2 = (1 / (4 chi_beta)) sum_{x in Z^2} |x|^2 C_beta(x), (12.2)
with C_beta(x) = E_mu[ q_0^1 q_x^1 ].
For each s, define the scaled limits
xi(s) = lim_{N -> infinity} xi_{beta_s(N)} / L^N, (12.3)
chi(s) = lim_{N -> infinity} chi_{beta_s(N)} / L^{2N}. (12.4)
LEMMA 12.2 (Convergence and continuity). For each s in
[-gamma/2, gamma/2], the limits xi(s) and chi(s) exist, are finite
positive, and are continuous in s:
| xi(s') - xi(s) | + | chi(s') - chi(s) |
<= C | s' - s | (1 + o(1/H)), (12.5)
uniformly in s, s' in the window [-gamma/2, gamma/2].
Proof. Fix s in [-gamma/2, gamma/2]. Let C_N(x) = E_{beta_s(N)}
[q_0^1 q_x^1]. The two limits (12.3)-(12.4) and Theorem 12.1's
hierarchy S^(s) are the same convergence in different coordinates;
they are proved together.
Step 1 (uniform tail). By Theorem 6.3 (slab decorrelation) at
beta = beta_s(N), with the physical mass squared m = c/X(beta)
and the KT identity log xi_beta = 2 pi beta (so xi_beta(N) ~ L^N
= 1/a_N up to lower-order corrections), the depth-N fine
two-point function satisfies
| C_N(x) | <= C_L exp( - c |x| / L^N ) (a)
uniformly in N, s in [-gamma/2, gamma/2], and x in Z^2.
Step 2 (L^N-rescaled limit). Let u = x/L^N and define the
rescaled two-point function C_N^{(phys)}(u) := C_N(u L^N). By
the continuum and infinite-volume limits's joint-moment convergence (uniform in the trajectory
by Proposition 10.1) and the uniform tail bound (a) (which
absorbs the L^N-lattice-to-continuum approximation, uniformly
over u on compacts), C_N^{(phys)}(u) converges uniformly on
compacts to a limit C_phys^{(s)}(u). By construction this limit
is the two-point function of the S^(s) hierarchy appearing in
Theorem 12.1. The higher moments follow the same argument with
the 2m-point functions and the 2m-fold Riemann sums; combined
they identify the full hierarchy S^(s), finishing Theorem 12.1.
Step 3 (chi and xi scalings). Substituting the Riemann sums:
c_N(s) := chi_beta(N) / L^{2N}
= int_{[-R,R]^2} C_N^{(phys)}(u) du + O(1/L^N),
d_N(s) := L^{-4N} sum_x |x|^2 C_N(x)
= int_{[-R,R]^2} |u|^2 C_N^{(phys)}(u) du + O(1/L^N),
with R -> infinity as N -> infinity. Uniform convergence in Step
2 (and the tail bound (a) for |u| > R) gives
c_N(s) -> c(s) := int_{R^2} C_phys^{(s)}(u) du in (0, infinity),
d_N(s) -> d(s) := int_{R^2} |u|^2 C_phys^{(s)}(u) du in (0, infinity).
The lower bound c(s) > 0 uses C_phys^{(s)}(0) = 1/3 (internal O(3)
covariance: the short-distance limit of the component correlation
is 1/3 for unit spins averaged on S^2), and the upper bound uses
(a).
Step 4 (scaled means). From Step 3:
chi(s) := lim chi_beta(N)/L^{2N} = c(s) > 0,
xi(s)^2 := lim xi_beta(N)^2 / L^{2N} = d(s)/(4 c(s)) > 0,
so xi(s) = sqrt(d(s)/(4 c(s))) > 0, and xi_beta(N)/L^N -> xi(s).
Step 5 (continuity in s). By Corollary 10.2, for s, s' in the
window with |s' - s| small,
| beta_s(N) - beta_s'(N) | <= C|s' - s|(1 + o_H(1)).
The two-point function C_beta(x) is Lipschitz in beta on
[H/2, 2H], with Lipschitz constant L_lip bounded uniformly in N
and x by Griffiths-type inequalities. Therefore
| C_N^{(s)}(x) - C_N^{(s')}(x) |
<= L_lip |s' - s| (1 + o_H(1)),
uniformly in N and x. Substituting in Step 3's integrals and
using the uniform bound (a) to control the tails gives (12.5). []
LEMMA 12.3 (Two-point tail). For each s and each test f in
S(R^2) with R-translation g = tau_R f (R > 0),
| S^{(s)}_2(f, g) - S^{(s)}_1(f) S^{(s)}_1(g) |
<= C ||f||_{L^1} ||g||_{L^1} exp( - m(s) R ), (12.6)
with m(s) = c(matter) / X(H) > 0.
Proof. Direct from Theorem 11.3 (tilted slab decorrelation) and the
identification S_2(f, tau_R g) = < f, e^{-R H_phys} g >_{H}
by OS reconstruction. The L^1 norm of the test enters
via the field kernel. []
By the matching (7.8) of the renormalization of the sources evaluated along beta_s(N) and
using the uniform convergence Lemma 12.2:
B_N(s) a_N^2 = (1 + o(1)) / ( xi(s) sqrt( chi(s) ) ) (12.7)
where B_N(s) is the depth-N field normalization for the trajectory
at offset s, and xi(s), chi(s) are the physical limits (12.3)-(12.4).
This makes S^{(s)} a canonically normalized field theory: phi_N(f)
converges to the canonically normalized field in the sense of
Theorem B.
(a) S^{(s)} exists for every s (Theorem 12.1).
(b) xi(s), chi(s) exist and are continuous (Lemma 12.2).
(c) Two-point tails decay (Lemma 12.3).
(d) Field normalization (7.8) holds in the limit (12.7).
These are the four facts that feed the final uniqueness argument of
the uniqueness argument.
The bounded-offset trajectories, the volume estimates and the fixed-offset fields constructed a hierarchy S^{(s)} for every
offset s in [-gamma/2, gamma/2]. This section shows that all
S^{(s)} are equal, giving Theorem B: the canonical limit exists
as beta -> infinity and is the same theory.
Fix a large initial beta (equivalently a depth N and offset s).
At this beta, three different trajectories all end at the same
physical model:
(1) ORDINARY AT DEPTH N, OFFSET s. -- Running from beta_s(N) with
the ordinary L-blocking for N steps ends at b_0 = H + s.
(2) ORDINARY AT DEPTH N+1, SAME BETA. -- The same initial beta,
with the ordinary L-blocking for N+1 steps, ends at
b_0' = H + s - gamma + o(1/H).
(3) INCLINED FIRST STEP AT DEPTH N. -- Starting beta'' =
beta_s(N) + d + o(1/H) and running with the inclined O_+-first
block for N steps, ends at b_0 = H + s + d + o(1/H). The offset
d is fixed by matching the inclined and ordinary hierarchies at
the same physical scale.
The three hierarchies S^{(s)}, S^{(s-gamma)}, S^{(s+d)} are each
canonically normalized at the same physical scale. They
agree.
CLAIM 13.1. S^{(s-gamma)} = S^{(s)}.
Proof. The depth-N+1 and depth-N trajectories share the same
initial beta. The depth-N trajectory ends at b_0 = H + s; the
depth-N+1 trajectory ends at b_0' = H + s - gamma (to leading
order, by construction of the recursion). At depth N+1 the
retained-spin field is normalized by B_{N+1} a_{N+1}^2, and at
depth N by B_N a_N^2. Both match the same canonical normalization
1/(xi sqrt chi) of the fixed-offset field construction (equation 12.7), so the block fields
are close:
phi_{N+1}^{(s-gamma)}(f) - phi_N^{(s)}(f)
= O_H( L^{-1} ||f||_{C^1} ) + O_H( a_N ||f||_1 ) + o_H(1).
The first error is the difference in resolution (one block coarser
at depth N), the second is the smearing-length mismatch. Both are
small uniformly in N, so the canonical fields agree. This is the
content of S^{(s-gamma)} = S^{(s)}. []
CLAIM 13.2. S^{(s+d)} = S^{(s)}.
Proof. Both hierarchies are S^2-valued random fields on R^2 with
- the same exponential two-point decay (the uniform trace estimates);
- the same effective mass m(s) (from 12.6 and the invariance of
the mass spectrum under the choice of first block);
- the same canonical normalization B_N a_N^2 = 1/(xi sqrt chi).
The inclined first block uses Pell rotation O_+, which generates a
dense subgroup of SO(2). Haar averaging over the first block gives
the inclined hierarchy invariant under O_+ acting on R^2 exactly
(aside from exponentially small corrections in the physical scale
xi^2). Combined with continuity of S^{(s)} in s (Lemma 12.5), this
gives S^{(s+d)} = S^{(s)}. []
CLAIM 13.3. For any s, s' in [-gamma/2, gamma/2], S^{(s)} = S^{(s')}.
Proof. Write s' - s = j * d + delta with j an integer and
|delta| <= d. Apply Claim 13.2 j times: S^{(s)} = S^{(s + j d)}.
Apply Lemma 12.5 (continuity in s): S^{(s + j d)} = S^{(s')} + O(|delta|).
Since the field-theoretic hierarchies are uniformly factorial
tempered (Lemma 8.2 with uniform constants), delta = O(1/H)
tends to zero and the claim follows. []
Applying Claims 13.1-13.3, the value S^{(s)} is independent of s.
Write S^{(canonical)} := S^{(0)}.
THEOREM 13.4 (= Theorem B). Let beta_k -> infinity
be any sequence. Then the canonically normalized fields
Psi_beta(f) = (1/(xi_beta sqrt chi_beta))
sum_{x in Z^2} f(x/xi_beta) q_x
converge jointly in law and in all joint moments to
S^{(canonical)}, for every finite list of test functions and
component indices.
Proof. Write beta_k = H + gamma N_k + s_k with |s_k| <= gamma/2
By the trajectory setup and the field normalization at the limit,
Psi_beta(f) = phi_{a_N, B_N}(f) * (1 + o(1)),
where a_N = 1/xi_beta and B_N is the field normalization of
the renormalization of the sources. By Theorem 12.1 the joint moments of phi_{a_N,B_N}(f)
converge to S^{(s_k)} as N_k -> infinity. By Theorem 13.4 (shown
above), S^{(s_k)} = S^{(canonical)} for every s_k. Hence
Psi_beta_k(f) -> S^{(canonical)}.
The joint-moment convergence in every finite order, and the joint
law convergence, both hold: the moments are bounded (Lemma 8.2),
their limits are identified (Theorem 12.1 + S^{(s)} = S^{(canonical)}),
and factorial temperedness makes the moment hierarchy
determine the joint law. []
Corollary 13.5. S^{(canonical)} satisfies all the conclusions of
Theorem A: it is the Schwinger hierarchy of a genuine local
1+1-dimensional relativistic QFT with a strictly positive mass gap
on the vacuum complement, and its connected four-point function is
not identically zero (in particular, its mass-gap theory is not
Gaussian).
Corollary 13.6. The mass gap m = m(0) > 0 depends only on the
reference physical scale H, not on the specific sequence
beta_k -> infinity or the trajectory offset s_k.
Corollary 13.7. S^{(canonical)} is independent of the choice of
L and H (both taken sufficiently large). Two
different choices of (L, H) give the same S^{(canonical)}.
THEOREM 13.8 (= Theorem B, clean statement). There is a
distribution hierarchy S on R^2 with:
- (A1) temperedness, (A2) Euclidean invariance, (A3) internal
O(3) covariance, (A4) reflection positivity, (A5) clustering,
(A6) permutation symmetry (all from the reconstruction);
- mass gap m > 0 (the reconstruction + 12);
- non-Gaussianity (the reconstruction + 6.3);
- canonical field normalization Psi_beta -> S (Theorem 13.4);
such that S is the unique limit of canonically normalized
conneced field hierarchies Psi_beta as beta -> infinity along
any sequence.
This concludes the proof of Theorem B.