Let E* be the set of unordered nearest-neighbor edges of Z^2. On a
probability space (Omega_env, P_env) let (xi_e), e in E*, be independent
signs with P_env(xi_e = +1) = P_env(xi_e = -1) = 1/2. Fix 0 < eps < 1 and
J_e(omega) = 1 + eps * xi_e(omega). (1.1)
Every bond is ferromagnetic (coupling 1+eps or 1-eps, always positive,
always preferring aligned neighbors). The SAME infinite environment is used
for all finite boxes. Smaller eps means weaker disorder.
For m >= 1 let Lambda_m = {-m,...,m}^2. With spins sigma in
{-1,+1}^{Lambda_m}, extended by +1 outside (the "+ boundary condition"),
H^+_{m,omega}(sigma) = - sum over edges e touching Lambda_m of
J_e(omega) sigma_x sigma_y . (1.2)
Write < . >^+_{m,beta,omega} for expectation under the normalized weight
exp(-beta H^+), normalized separately for each environment.
For each environment the limit <sigma_0>^+_{infty,beta,omega} exists by
monotonicity in m. Define the quenched magnetization and the critical
inverse temperature by
m_eps(beta) = E_env[ lim_{m->infty} <sigma_0>^+_{m,beta,omega} ],
beta_c(eps) = inf{ beta >= 0 : m_eps(beta) > 0 }. (1.3)
This is the definition proved to be correct. No assumption is made
that self-duality identifies the critical point; self-duality is only
a bookkeeping device.
Since 1-eps <= J_e <= 1+eps, comparing to the clean model with constant
coupling gives, with K0 = (1/2) log(1+sqrt2),
0 < K0/(1+eps) <= beta_c(eps) <= K0/(1-eps) < infinity. (1.4)
So the critical point is finite and positive; the only questions are its
exact location and the interface law AT it.
The interface is the set of dual edges separating + from - spins. This becomes
a bijection so that "spin law" and "curve law" are literally the same
object.
Fix a finite V subset Z^2, let Q_V = union over v in V of the closed unit
square centered at v, D_V = interior(Q_V). Assume Q_V is the closure of a
Jordan domain (a nice simply connected region with no pinches). Let E_V be
the dual edges crossing primal bonds with at least one endpoint in V. Pick
two distinct marks f_in, f_out on the boundary in the dual lattice, and
attach to each a short exterior stub e_in, e_out lying outside Q_V.
For an edge set A and dual vertex f, let d_f(A) be its degree. The
admissible interior contours are
P_V = { P subset E_V : for every f, d_f(P) + 1_{f=f_in} + 1_{f=f_out}
is even }. (2.1)
P_V is required to be nonempty and to admit a finite completion P_o
outside E_V containing the two stubs, with the same two odd marks. Call all
this "admissible lattice data"; P_o is fixed deterministically.
For a fixed environment at inverse temperature beta the contour law is
mu_{V,beta,eps,omega}(P)
= (1/Z) exp( -2 beta sum_{e in P} J_{e*}(omega) ), P in P_V. (2.2)
LEMMA 2.1 (Contour = spin image).
The law (2.2) is the exact image of the nearest-neighbor Ising law on V with
the exterior spins fixed by P_o, and it does not depend on the choice of P_o.
Proof. An even finite dual edge set determines a unique spin configuration
equal to +1 far away: the sign at a primal vertex is the parity of crossings
of a path to infinity, well defined exactly because the set is even.
Restricting a contour to E_V is a bijection between compatible spin
configurations on V and P_V. Across a primal bond the spin product is -1
exactly when its crossing dual edge is in the contour. Hence
exp( beta sum_{edges touching V} J sigma_x sigma_y )
= exp( beta sum_{e in E_V} J_{e*} ) * exp( -2 beta sum_{e in P} J_{e*} ).
The first factor is constant in P and every exterior-only energy is
constant, so after normalization this gives (2.2), which visibly does not
involve P_o. QED.
Add to P the half of each stub from its mark to its midpoint. At degree-two
dual vertices join the two occupied edges; at degree-four dual vertices pair
north with east and south with west (and the same at the marks). Draw these
pairings disjointly in tiny disks of radius < 1/10. The resulting simple
oriented curve from the incoming stub midpoint to the outgoing one is
gamma(P).
3. THE LIMITING LAW: CHORDAL SLE3
For continuous oriented curves gamma, eta represented on [0,1] define
d_curv(gamma,eta) = inf over increasing homeomorphisms alpha,rho
of sup_{t in [0,1]} |gamma(alpha(t))
- eta(rho(t))|.
Curves at distance zero are identified. This is uniform distance modulo
reparametrization: it sees shape and orientation, not speed. Let
dbar = min{1, d_curv}. For two probability measures nu, nu' on this space,
d_BL(nu,nu') = sup over F with |F|<=1 and |F(g)-F(h)| <= dbar(g,h)
of | integral F d nu - integral F d nu' | . (3.1)
Convergence in d_BL is exactly weak convergence tested by bounded functions
that are Lipschitz for the curve metric.
Let D be a bounded Jordan domain and a,b distinct boundary points. Chordal
SLE_3(D;a,b) is the law obtained by conformally mapping to D the
upper-half-plane Loewner trace
d/dt g_t(z) = 2 / ( g_t(z) - sqrt(3) B_t ), g_0(z) = z, (3.2)
with B a standard real Brownian motion, sending 0 to a and infinity to b.
The oriented curve is kept modulo reparametrization. (It is the canonical
random curve with conformal symmetry; sqrt3 is what makes it the Ising
interface law.)
THEOREM 4.1 (Quenched interface universality).
There is eps_0 > 0 such that for every fixed eps in (0, eps_0) the following
holds. Let D be a bounded Jordan domain and a,b distinct boundary points.
Take any deterministic sequence of admissible lattice data with meshes
delta_n -> 0 and boundary homeomorphisms
phi_n : boundary(D) -> boundary(delta_n D_{V_n}),
sup_z |phi_n(z) - z| -> 0, phi_n(a)=delta_n f_in,n,
phi_n(b)=delta_n f_out,n,
i.e. the discrete domains converge to D with the two marks going to a,b.
Let Q_{n,omega} be the law of delta_n gamma(P) under
mu_{V_n, beta_c(eps), eps, omega} from (2.2). Then for every eta > 0,
P_env[ d_BL( Q_{n,omega}, SLE_3(D;a,b) ) > eta ] -> 0. (4.1)
That is: for a typical fixed environment, the rescaled interface law
converges, in the full oriented-curve topology, to SLE3. The convergence is
quenched: the randomness is the disorder, not an average over it.
The proof pins the critical temperature. For sufficiently small eps,
( e^{2 beta_c(eps)(1+eps)} - 1 )( e^{2 beta_c(eps)(1-eps)} - 1 ) = 2. (4.2)
This is the self-dual tuning, but (4.2) is DERIVED from the magnetization
threshold in Section 10; it is not assumed.
All scale parameters and the disorder interval (0, eps_0) are chosen BEFORE
the domain D and the approximations are fixed. eps is fixed as the mesh -> 0.
The domain approximations do not depend on the environment, and make use of their
original Euclidean embedding throughout.
This section is the map. Read it now; come back to it whenever a later
section feels aimless. Each item below is one section of the proof.
A single number is sought, attached to each local observable, measuring "how
much pure energy it contains", and that number is required to be EXACTLY
preserved when the observable is averaged through a clean annulus. The clean
energy covariance already fixes the energy direction; a linear
functional Q is then calibrated on it. Two consequences are needed later: for two far-apart
centered observables the energy charge of their product nearly cancels
(fusion), and a large positive local weight, when averaged inside a square,
has small oscillation across the boundary.
At spacing s, rescale to spacing Ls. Because the interactions must be kept
local in the ENVIRONMENT (a fixed realization, not an average), fresh
auxiliary coins are added, separated squares are selected with probability p,
their interactions are pushed to square boundaries via normalized coin
likelihoods, and rare large terms are handled by isolated concentrations. The output is again a family of
bounded-range centered even potentials. The map is exactly reversible: two
consecutive scales can be coupled so they differ only on small squares, with
independent marks whose parameters are small.
Four numbers are tracked of a scale-s field: X (size), Y (environmental mean),
m (mean charge), W (the variance density of the charge). One step nearly
multiplies each by a contraction factor, leaving only the (uncontracted) W as
a source in the bound for X.
W itself is replaced by a polynomial coordinate v_H whose quadratic part is W.
One scale step changes W by a NEGATIVE multiple of W^2 (the normalization of a
joint insertion of two separated interactions: its denominator subtracts at
fourth order). The sign of the flow comes from tiny cancellations.
Using a shooting argument on the initial temperature offset h, the
expanding charge mean is kept inside walls m = +/- Dv while v obeys v_{k+1} <=
1/(v_k^{-1}+c). Thus v_k ~ 1/(c k): the disorder decays like 1/k. That decay
is NOT summable.
A change in one small rescaled square can alter a fixed macroscopic test only
if many contour passages reach out of that square. In the clean model each such
passage has probability a positive power of the size ratio. Summing these
geometrically weighted errors over scales makes the total comparison tend to
zero, even though the individual errors are not summable. An exterior supremum is taken
supremum first, so distinct ring neighborhoods multiply.
Torus duality shows the tuned parameter equals zero, i.e. the self-dual point,
but the real content is a magnetization argument: the tuned coupling is
exactly the threshold for plus magnetization. Decision-tree inequalities and
long independent annuli give the needed sharpness.
Boundary barriers (signed strip crossings) couple the target interface to pure
comparison chords whose law is deterministic even after conditioning on the
environment. The expected distance in these couplings bounds the expected
bounded-Lipschitz distance between the quenched law and SLE3. A multiple-
passage exclusion kills backtracking. This proves Theorem 4.1.
Everything in this section is about the CLEAN critical Ising model: no
disorder. Its role is to provide a bookkeeping coordinate that the random
rescaling will preserve exactly.
Let mu_0 be the full-plane nearest-neighbor Ising state at the clean critical
coupling K0 = (1/2) log(1+sqrt2). Write E_0 for its expectation. Let
B_s(x) = x + [-s,s]^2 intersect Z^2, and dB_s(x) = B_s(x) minus B_{s-1}(x).
Abbreviate B_s = B_s(0).
A local observable is EVEN if it is unchanged under flipping all spins. It
may depend on independent auxiliary variables attached to its support; those
are integrated first, and spin flip leaves them alone. All norms are sup
norms; centered means E_0-mean zero (including auxiliary integration).
Condensation of two spins. On a nearest-neighbor edge {z,z'}, critical duality
gives E_0 sigma_z sigma_z' = coth(2K0) - E_0 sigma_z sigma_z'. Since
coth(2K0) = sqrt2, this gives E_0 sigma_z sigma_z' = 1/sqrt2. (Check:
coth(2K0) = (e^{2K0}+e^{-2K0})/(e^{2K0}-e^{-2K0}) = (2sqrt2)/2 = sqrt2.)
So define the centered bond energy, with kappa = 1/pi^2:
b_e = sigma_x sigma_y - 1/sqrt2 (for edge e = {x,y}),
B_x = (1/4) sum_{e containing x} b_e, B = B_0. (6.1)
b_e has clean mean zero. B_x is the local energy density at site x. kappa is the
exact coefficient of the energy covariance (item 6.2).
Transfer operator. For an observable strictly inside B_R(x), define
P_{R,x} f = E_0[ f | sigma on dB_R(x) ] - E_0 f.
P_{R,x} "averages out the interior, fixes the boundary". Spatial Markov gives
exact composition for nested squares, and permits integration inside disjoint
squares while keeping all correlations with their exterior.
(I1) BUFFERED COMPARISON. On one side of a fixed proportional unpinned
collar, under any data on the other side, the spin-law density is between
K^{-1} and K times the full-plane marginal, with K depending only on the
collar ratios. (So conditional expectation through such a collar is
bounded, and this holds for mixtures of boundary data and interchanged
sides.)
(I2) ENERGY COVARIANCE. For either orientation of each bond,
Cov_0(b_e, b_{e'+x}) = (kappa + o(1))/|x|^2 as |x| -> inf. (6.2)
Same for B, B_x; fixed displacements do not change the coefficient.
(I3) BOUNDED REPLACEMENT + RADIAL SPECTRAL GAP. Bounded lattice observables on
finitely many positively separated square/band supports have pairings
approximated simultaneously by bounded continuum spin-field tests, with
the same sup bounds; spatial smears are fixed before the mesh limit. The
radial dilation semigroup on even quarter-turn-invariant tests has, for
spectrum below 3, only a constant state at 0 and one state at 1. Lattice
smears Phi_s(psi) = a_sigma^{-1} s^{-15/8} sum_z psi(z/s) sigma_z have
convergent moments of all orders, uniform in s; the two-point clean
correlator is S(z,z') = |z-z'|^{-1/4}.
Quarter-turn averaging about the center of B_s is denoted A_s; "ker A_s" means
the AVERAGE of the observable vanishes (not each value).
This lemma is the engine of the argument. It says: transferring an observable
from scale s to scale ds through the clean model kills two things precisely -
it shrinks the size by d^{-1} in the energy direction, and by d^{-3} away from
it; and on observables with zero quarter-turn average it does better, d^{-2}.
LEMMA 6.1 (Fixed-ratio pure transfer).
There is a constant C with the following property. Fix finitely many
sufficiently large integer ratios d. A common measurement ratio H, then a
threshold s_*, can be chosen so that for every integer s >= s_* there are
linear functionals l_s on centered even quarter-turn-invariant observables in
B_s with ||l_s|| <= C and centered right inverses e_s with l_s(e_s) = 1,
||e_s|| <= C, and for each chosen ratio d, with T_{s,d} = P_{ds},
||T_{s,d} f|| <= C d^{-1} |l_s(f)| + C d^{-3} ||f||, (6.3)
|l_{ds}(T_{s,d} f) - d^{-1} l_s(f)| <= C d^{-3} ||f||. (6.4)
If f is even and A_s f = 0, then ||T_{s,d} f|| <= C d^{-2} ||f||. The constant
C is independent of the chosen large ratios.
Proof. On the invariant subspace use the upper-limit estimates of [S, Sec 3,
"Compatible lattice coefficients and arbitrary boundary data"], dividing the
companion's density-normalized operator by d^2. The measured coefficient is
l_s(f) = (H g) Cov_0(f, W_{Hs}),
where W is the fixed exterior annular test there and g (nonzero) is its radial
degree-one coefficient. The remainder in the transfer norm is
C d^{-3} + C/(d H^2), and in coefficient compatibility it is C/(d H^2). Choose
H large enough for all prescribed ratios, then the replacement accuracies, and
finally s_* to absorb the uniform upper-limit errors into C d^{-3}. Center the
companion's fixed interior reference test under E_0 and divide by its measured
coefficient to get the centered bounded right inverse; centering changes neither
that coefficient nor the transfer estimates. The radial spectral gap needs only
quarter turns, so no reflection averaging is required.
For the nonscalar bound, the same method works. Before the angular averaging in
the radial gap, the polynomial expansion is
(F, D_{-log r} G)_rad = E conj(F) E G
+ r conj(c(F)) c(G) + O_{F,G}(r^2),
with both displayed coefficients rotation scalars. Before smearing,
c(prod_i sigma(z_i)) = S(z_1,...,z_{2m}) A_2(z)/A_0(z),
A_q(z) = sum over u_i in {-1,1} with sum u_i = q of
prod_{i<j} |z_i-z_j|^{u_i u_j / 2}.
All distances are rotation invariant, so both vanish on the even
zero-quarter-turn-average subspace. A polynomial self-pairing there is an
integral of r^a against a positive spectral measure and is O_F(r^2); positive
mass in any compact subinterval of [0,2) would contradict this as r->0. So the
whole spectrum of this subspace sits in [2,inf), giving the uniform bound
r^2 ||F||_2 ||G||_2. Apply bounded replacement (I3) to the observable and to the
buffered boundary density, project the first proxy with I - A (changing norm and
replacement error by at most a factor two), and the radial bound gives C d^{-2}
uniformly over the boundary assignment. The same finite-list choice of
accuracies and threshold makes the lattice inequality uniform in s and in the
observable. This proves the lemma without any spatial symmetry of the original
even observable. QED.
An EXACTLY preserved linear functional is sought, not just the approximate l_s of
Lemma 6.1. Fix u = 9/5 and w = 12/5 (these exponents are the savings this gives
below). Fix a large ratio d and the geometric scale sequence s_j = s_* d^j.
LEMMA 6.2 (Stable hyperplane).
With d large enough, on centered even quarter-turn-invariant observables in
B_{s_j} there are uniformly bounded linear functionals beta_j such that
|beta_j(f) - l_{s_j}(f)| <= C d^{-2} ||f||,
beta_{j+1}(P_{s_{j+1}} f) = m_j beta_j(f), m_j = d^{-1}(1 + O(d^{-2})) > 0.
An observable in ker beta_j satisfies, for all n >= 1,
||P_{s_{j+n}} f|| <= C d^{-w n} ||f||. (6.5)
Every observable outside this kernel eventually enters an invariant cone
||f|| <= M |l_{s_j}(f)|, (6.6)
with M independent of j and of d (large enough).
Proof. Start with a bounded right inverse v_0 of l_{s_*} and propagate it:
m_j = l_{s_{j+1}}(P_{s_{j+1}} v_j), v_{j+1} = m_j^{-1} P_{s_{j+1}} v_j.
Lemma 6.1 shows by induction that ||v_j|| stays bounded and
m_j = d^{-1}(1+O(d^{-2})). Concretely, take a fixed bound larger than four
times the constant in (6.3), then take d so large that the d^{-2} errors keep
it. Write f_j = a_j v_j + k_j with l_{s_j}(k_j) = 0. Transfer is triangular,
a_{j+1} = m_j a_j + t_j k_j, k_{j+1} = K_j k_j,
||t_j|| + ||K_j|| <= C d^{-3},
the last by subtracting v_{j+1} l_{s_{j+1}}(P k_j) from the transferred kernel
vector. Put M_{j,n} = prod_{h=j}^{n-1} m_h (with M_{j,j}=1) and define
beta_j(a v_j + k) = a + sum_{n=j}^{inf}
( t_n K_{n-1}...K_j k ) / M_{j,n+1}.
For n=j the K-product is the identity. The series converges in operator norm
since successive bounds differ by O(d^{-2}); its correction to a has norm
O(d^{-2})||k||. Shifting the series gives exact compatibility with multiplier
m_j.
If beta_j(f)=0, the scalar at each later time equals minus the remaining tail,
so has magnitude at most C d^{-2}||k_n||; hence the whole vector contracts at
rate (C d^{-3})^{n-j}. Enlarge d so this is <= d^{-w(n-j)}. If beta_j(f) is
nonzero, dividing by M_{j,n} shows the scalar converges to beta_j(f) (nonzero)
while the kernel part tends to zero, so it enters the cone. For a vector in the
cone,
||P_{ds} f|| / |l_{ds}(P_{ds} f)|
<= (C + C d^{-2} M)/(1 - C d^{-2} M);
choose M larger than the limiting bound, then d large. This keeps the cone
invariant. QED.
The whole energy direction lives in P_s B. Its size is required.
LEMMA 6.3 (Energy calibration).
For all sufficiently large integer s,
c/s <= ||P_s B|| <= C/s, (6.7)
and along the geometric scales, P_s B eventually belongs to the cone (6.6).
Proof. If a transferred B lay in the stable hyperplane, then (6.5) plus the
Markov property, tested against a distant B_x, would give
E_0[B B_x] = O(|x|^{-w}). But w = 12/5 > 2, contradicting the energy
covariance (6.2). So P_s B is not stable and the cone statement follows from
Lemma 6.2.
Next, one sufficiently large fixed T makes any centered unit-norm cone
observable f_s satisfy, uniformly in large s,
E_0[ f_s (tau_{T s e_1} f_s) ] >= c_T > 0, (6.8)
with tau_x the translation of an observable by x, and c_T of order T^{-2}.
Take T divisible by 8. Choose nonzero nonnegative smooth bumps psi_1, psi_2
in two disjoint regions strictly inside the unit square, and let Y_s be the
quarter-turn average of Phi_s(psi_1) Phi_s(psi_2), centered. Its continuum
counterpart, averaged under quarter turns and centered, has nonzero scalar
coefficient. For two unsmeared pairs z_1,z_2 and T e_1 + y_1, T e_1 + y_2 the
squared four-spin formula of [S] gives
Cov(sigma(z_1)sigma(z_2), sigma(Te_1+y_1)sigma(Te_1+y_2))
= |z_1-z_2|^{3/4} |y_1-y_2|^{3/4} / (4 T^2) + o(T^{-2}).
Neutral within-pair assignments give the product term; cross logarithms are
O(T^{-2}) and cancel to first order after sign summation. The charge +2/-2
assignments give the displayed strictly positive coefficient after the positive
square root. Uniform on the separated bump supports, so it survives smearing
and averaging.
Take a large fixed A and set V_s = chi_A(Y_s) - E_0 chi_A(Y_s), chi_A the clamp
to [-A,A]. The nonzero scalar coefficient survives for all large A: moment
bounds give L^2 convergence of the truncated continuum test, and the radial
coefficient is continuous in that norm; its measured lattice coefficient is
then bounded away from zero once H and the threshold are large. To see A may be
fixed before T, put eps_A = sup_s E_0|Y_s - V_s| -> 0 (moment bounds). Buffered
comparison gives
||P_{2s}(Y_s - V_s)|| <= K eps_A, ||P_{2s}Y_s|| + ||P_{2s}V_s|| <= C,
the last independent of A since E_0|V_s| <= 2 E_0|Y_s|. Apply the C/T transfer
bound of Lemma 6.1 to both factors conditioned on these radius-2s collars, and
integrate in the two disjoint radius-Ts/4 squares:
|E_0[Y_s tau Y_s] - E_0[V_s tau V_s]| <= C T^{-2} eps_A.
So a fixed large A preserves the positive T^{-2} coefficient, and for large
fixed T the covariance of V_s is at least cT^{-2} once the lattice threshold is
large.
Now put a_s = l_s(f_s)/l_s(V_s); the cone condition and coefficient bounds give
0 < c <= |a_s| <= C. The observable f_s - a_s V_s has measured coefficient zero.
Transfer it and the other factor into the disjoint radius-Ts/4 squares:
Lemma 6.1 bounds the first by C T^{-3} and the other centered factor by
C T^{-1}. Thus
E_0[f_s tau f_s] = a_s^2 E_0[V_s tau V_s] + O(T^{-4}),
proving (6.8). Choose the truncation, then T, then a common measurement distance
for d, T/4, T/8, and finally the lattice threshold (T/8 used after transferring
a truncation error to its radius-2s collar). Lemma 6.2 is reapplied with these
choices; its constants stay uniform. For this final choice the covariance
contradiction again shows P_s B is not stable, so it enters the newly built
cone.
Apply (6.8) to f_s = P_s B / ||P_s B||. By simultaneous Markov integration in
the two disjoint squares,
E_0[(P_s B) tau (P_s B)] = E_0[B B_{T s e_1}].
The cone lower bound and (6.2) give the upper bound C/s; the bound
|E_0[f_s tau f_s]| <= 1 gives the lower bound c/s. These hold along the
geometric scales; for an intermediate radius the tower property squeezes its
norm between the norms at the two adjacent scales, giving (6.7) for every large
s. QED.
The distinguished functional is now assembled. It assigns to each local
observable its "pure energy content", is exactly preserved by clean transfer,
and controls how much of an observable survives being averaged out.
THEOREM 6.4 (Exact pure energy charge).
For every bounded even local observable f the limit
Q(f) = kappa^{-1} lim_{|x|->inf} |x|^2 E_0[ f B_x ] (6.9)
exists. It is linear, kills constants, is invariant under lattice
translations and square symmetries, and satisfies Q(b_e) = 1 for every edge.
Pure transfer preserves Q exactly. If f is supported in B_s(x) then
|Q(f)| <= C s ||f||. (6.10)
For centered f and R >= 4s,
||P_{R,x} f|| <= C (s/R) ||f||. (6.11)
If Q(f) = 0 the factor s/R improves to (s/R)^u; it improves to (s/R)^w if,
after auxiliary integration, f is also quarter-turn invariant about x. In
general, with v = u or w according to these two symmetry classes,
||P_{R,x} f - Q(f) P_{R,x} B_x|| <= C (s/R)^v ||f||. (6.12)
All constants are independent of s, R, f.
Proof. Work first at the origin, on the geometric scales s_j = s_* d^j, with
quarter-turn-invariant centered functions. By Lemma 6.3 the vector
E_j = P_{s_j} B lies eventually in the cone and has norm comparable to 1/s_j.
Since beta_j = l_{s_j} + O(d^{-2}) and d is large,
c/s_j <= |beta_j(E_j)| <= C/s_j.
Define, temporarily, tildeQ(f) = beta_j(f)/beta_j(E_j) for f supported in
B_{s_j}. Exact compatibility of numerator and denominator makes this invariant
under transfer to later geometric scales. It is also consistent when the same
function is regarded as living in a later support space: the difference between
it and its transfer vanishes after the next transfer, hence has zero beta by
compatibility. Its kernel is exactly the stable hyperplane, and
|tildeQ(f)| <= C s_j ||f||.
Increase the fixed starting scale so calibration holds on the whole grid. For
f supported in radius s above that threshold, use the next containing
geometric scale r with s <= r <= ds. Compare f with tildeQ(f) P_r B, whose
norm is at most C ||f|| by (6.7). Their difference is stable, so Lemma 6.2
followed by norm contraction between adjacent scales proves (6.12) with v = w
and tildeQ in place of Q. Radii R within a fixed factor of s are absorbed by
increasing the constant.
For a general even function, apply the construction to its quarter-turn
average and set tildeQ(f) = tildeQ(A_s f). The complementary component
contracts at rate (C d^{-2})^j by the nonscalar bound in Lemma 6.1. Choose d so
that this is at most d^{-u j}. This gives the general remainder estimate with
v = u. Together with ||P_R B|| <= C/R, it gives the even-transfer bound (6.11).
For fixed f, pair its remainder with B_x and integrate both factors in disjoint
squares of radii proportional to |x|. The remainder has norm O_f(|x|^{-u}) and
the transferred B_x has norm O(|x|^{-1}). Since u = 9/5 > 1, their covariance
is o(|x|^{-2}). Hence (6.2) gives
|x|^2 E_0[ f B_x ] -> kappa tildeQ(f).
This proves existence in (6.9) and identifies Q = tildeQ. So the definition is
independent of the geometric grid and its starting scale. Translation and
square invariance follow from this intrinsic limit and the corresponding
symmetries of mu_0; the limit is unchanged by a fixed displacement of its
distant test. Equation (6.2) gives Q(b_e) = 1 for every edge. The Markov
property preserves the covariance with every sufficiently distant B_x, hence
preserves Q under transfer through any containing square. This also handles
arbitrary centers: the finitely many radii below the fixed threshold are
absorbed by transferring first to that threshold and enlarging constants.
Integrating independent auxiliary variables at the outset extends everything to
them. QED.
Write q_s(f) = Q(f)/s. Two centered energy observables, far apart, have a
product whose energy charge is much smaller than either factor's; Kramers-
Wannier duality supplies the cancellation.
PROPOSITION 6.5 (Covariance and charge of separated products).
Fix D < inf. Let f, g be centered even observables of norm at most one,
supported in B_{Ds}(x) and B_{Ds}(y), with s >= 1 and |x-y| = l s. For l large,
uniformly in s, x, y and the observables,
E_0[ f g ] = l^{-2} ( kappa q_s(f) q_s(g) + o_D(1) ), (6.13)
| q_s(f g) | <= C_D l^{-u}. (6.14)
If each observable is quarter-turn invariant about its own center after
auxiliary integration, (6.14) improves to C_D l^{-w}. The o_D(1) tends to zero
as l -> inf independently of s.
Proof. Put r = |x-y| and integrate in disjoint squares of radius a small fixed
fraction of r. Let F, G be the transferred observables and U_x, U_y the
corresponding transfers of B_x, B_y. Theorem 6.4 gives
||F|| + ||G|| <= C_D l^{-1},
||F - Q(f) U_x|| + ||G - Q(g) U_y|| <= C_D l^{-v},
with v = u, or w in the symmetric case. Note these compare with the bond
observable AFTER transfer to a containing scale; the untransferred Q(f) B_x can
have norm of order s. Hence
| E_0[fg] - Q(f) Q(g) E_0[B_x B_y] | <= C_D l^{-v-1}. (6.15)
Since v > 1, (6.2) proves (6.13); its error is uniform because r = l s >= l,
the energy remainder tends uniformly to zero outside growing disks, and
|q_s(f)| + |q_s(g)| <= C_D.
Simultaneous integration in the disjoint squares also preserves the charge of
the product: it preserves its correlation with every sufficiently distant
energy test, so (6.9) applies. All transferred products fit in a square of
radius C r. The charge bound applied to the same product difference as (6.15)
gives
(1/s) | Q(fg) - Q(f) Q(g) Q(B_x B_y) | <= C_D (r/s) l^{-v-1}
= C_D l^{-v}. (6.16)
It remains to prove
|Q(B_x B_y)| <= C r^{-w}. (6.17)
For distinct edges, Kramers-Wannier duality at criticality gives
E_0[ prod_{i=1}^n b_{e_i} ] = (-1)^n E_0[ prod_{i=1}^n b_{e_i^*} ], (6.18)
with e_i^* the crossing dual edge. To check the centering directly,
differentiate the finite planar partition-function duality [KW] in the distinct
couplings. Its coupling-dependent prefactor is prod_e (sinh(2K_e))^{1/2}, and
sinh(2K_e) sinh(2K_e^*) = 1. At K_0 the logarithmic prefactor derivative is
sqrt2 and dK_e^*/dK_e = -1; thus the operator (d/dK_e - 1/sqrt2) becomes
-(d/dK_e^* - 1/sqrt2). Distinct derivatives introduce no contact terms.
Exhausting the plane and using uniqueness of its pure critical state proves
(6.18).
Apply (6.18) to the three separated bond averages in Q(B_x B_y). After
identifying the dual lattice with the original one, the distant dual average
B'_z is centered and has charge one; so B'_z - B_z is centered with charge zero.
With x, y fixed, separately transfer the centered product
B'_x B'_y - E_0(B'_x B'_y) and this distant difference into squares of radii
proportional to |z|. Their norms are O_{x,y}(|z|^{-1}) and O(|z|^{-u}) by
Theorem 6.4, so their covariance is O_{x,y}(|z|^{-u-1}) = o(|z|^{-2}).
Replacing the distant dual average by B_z leaves the charge limit unchanged, so
Q(B_x B_y) = -Q(B'_x B'_y),
where the primed average uses the four dual edges surrounding the site. There
are four identifications given by the translations h = (+-1/2, +-1/2), sending
the dual average about x to the boundary-edge averages of the four unit
plaquettes incident to x. Let B'_{i,x}, i = 1..4, be the results and put
C_x = (1/4) sum_{i=1}^4 B'_{i,x}, D_{i,x} = B'_{i,x} - C_x.
C_x is quarter-turn invariant about x, has charge one, and is supported in a
fixed-radius square. Each D_{i,x} has charge zero and their average vanishes.
Hence
(1/4) sum_i Q(B'_{i,x} B'_{i,y})
= Q(C_x C_y) + (1/4) sum_i Q(D_{i,x} D_{i,y}).
The charge bound after separate transfers gives
|Q(D_{i,x} D_{i,y})| <= C r^{1-2u}, |Q(C_x C_y) - Q(B_x B_y)| <= C r^{-w}.
For the first, both factors have zero charge and decay as r^{-u}. For the
second, expand the difference with one factor C - B (quarter-turn invariant and
zero charge) while the other centered factor decays as r^{-1}; the
common-support charge bound supplies the factor C r. Averaging the negative
duality identity now gives 2|Q(B_x B_y)| <= C(r^{-w} + r^{1-2u}), and since
2u - 1 = 13/5 > 12/5 = w, this is (6.17).
Finally its contribution in (6.16) is at most
(|Q(f) Q(g)|/s) C r^{-w} <= C_D s (l s)^{-w}
= C_D s^{1-w} l^{-w} <= C_D l^{-w},
since w > 1. This proves both charge bounds uniformly for s >= 1. QED.
This is the clean input that permits treatment of "rare large terms" later: when a
positive weight sits inside a small patch, its effect on the partition function
does not depend much on the boundary around a bigger square.
LEMMA 6.6 (Transfer of a positive local weight).
There is C < inf with the following property. Let m, R be integers, m >= 1,
R >= 8m. Let eta be a finite family of independent auxiliary variables with
arbitrary reference laws, independent of the full-plane pure critical Ising
field. Suppose G = G(sigma_{B_m}, eta) is bounded, measurable, and even:
G(-sigma, eta) = G(sigma, eta). Write
||G|| = max_{sigma_{B_m}} ess sup_eta |G(sigma_{B_m}, eta)|,
Z_R^G(tau) = E_0[ e^G | sigma_{dB_R} = tau ],
with E_0 including the auxiliary integration. Then
osc_tau log Z_R^G(tau) <= C (m/R) min{1, ||G||}, (6.19)
where osc is sup minus inf over all boundary spin assignments, and C is
independent of the number and laws of the auxiliary variables. For a patch of
real radius a >= 0, take m = max{1, ceil(a)}.
Proof. Integrate the auxiliary variables and put
W(sigma_{B_m}) = E_eta e^G, Z = E_0 W,
h(xi) = E_0[ W | sigma_{dB_{2m}} = xi ] / Z.
W is positive and even. Buffered comparison (I1), applied to B_m and dB_{2m},
gives K >= 1 with
K^{-1} <= h(xi) <= K for every xi.
Indeed the conditional law on B_m has pointwise density between K^{-1} and K
relative to its full-plane marginal, which is a mixture of the same conditional
laws; integrating the positive W proves the bounds regardless of its size.
Spin-flip symmetry gives h(-xi) = h(xi), and the tower property gives
E_0 h = 1. If t = ||G||, then W and Z both lie in [e^{-t}, e^t], so
||h - 1|| <= min{ K + 1, e^{2t} - 1 } <= C_K min{1, t}.
Apply the even transfer estimate (6.11) of Theorem 6.4 to the centered even
function h - 1, supported in B_{2m}. Since R >= 8m,
||P_R(h - 1)|| <= C (m/R) min{1, t}.
The Ising Markov property, then conditional expectation, gives
Z_R^G(tau)/Z = E_0[ h | sigma_{dB_R} = tau ]
= 1 + P_R(h - 1)(tau) >= K^{-1}.
The logarithm is K-Lipschitz on [K^{-1}, inf), so its oscillation is at most
2K ||P_R(h - 1)||, which proves (6.19). QED.
Three tools are kept, all in the clean model and all exact:
(a) The charge Q (Theorem 6.4), exact under clean transfer, with the
remainder bound (6.12) governed by s/R to a power u or w.
(b) Fusion (Proposition 6.5): a product of two far-apart centered pieces has
energy charge only C_D l^{-u} (or l^{-w} when symmetric), with the
covariance behaving like l^{-2}(kappa q q + o).
(c) Positive-weight control (Lemma 6.6): a large local weight's average has
boundary oscillation C(m/R)(small).
These are precisely the ingredients the random rescaling in the next section
must preserve or bound.
The machine that carries the model from spacing s to spacing Ls is now built.
It must do three things at once: (1) keep every interaction local in the FIXED
environment (so disjoint regions stay independent); (2) preserve the exterior
marginals exactly, so consecutive scales can be coupled; (3) be symmetric in
law under translations and square symmetries, so stationarity survives. The
trick is a bank of fresh auxiliary fair coins, which belong to the Gibbs field
(not the environment), used to select well-separated squares to integrate.
Fix an integer spacing s. A cell has center s i, i in Z^2. Besides the spins,
the reference probability space may contain a locally finite family of
independent finite-valued auxiliary variables with specified plane locations.
Write E_0 for the pure critical Ising expectation TOGETHER with these reference
variables: they are part of the Gibbs field and are integrated or conditioned
along with the spins. In contrast, E is expectation over the bond environment
AND the independent choices (priorities, thresholds) used to build the
transformation; those are fixed when a Gibbs law is sampled. The product
reference law of the auxiliary variables is preserved by the grid translations
and square symmetries acting on their locations and labels; the fresh fair
comparison coins below share this property.
The input is a family of random functions f_i of spins and reference variables.
For each environment, f_i is real, bounded, spin-flip even, and centered,
E_0 f_i = 0. Its field support and the environment variables it depends on lie
within Cs of s i, in sup norm, for a fixed large numerical C. Primitive
environment variables are independent, so collections of f_i with disjoint
dependence neighborhoods are independent random functions. Invariance is assumed
in law under s-grid translations and square symmetries, acting also on the
auxiliary variables. Individual environments need not have these symmetries.
In a finite volume the logarithmic addition to the reference density is
sum_i f_i, over potentials meeting that volume, with old exterior field
variables prescribed. Every nontrivially decorated fresh coin is sampled as
part of the construction. Exactness holds for each fixed environment and
arbitrary old exterior assignment; the local implementation keeps complete
decoration stencils inside the free chart. It does NOT assert the same input
kernel after independently pinning some of those fresh coins. The construction
also works on a large torus. Infinite arrays only define local formulas and
their distributions; a sequential sampler is needed only in a finite volume.
Choose a large odd integer L and put s' = L s. The children of parent j are
C_j = L j + {-(L-1)/2,...,(L-1)/2}^2, F_j = sum_{i in C_j} f_i.
For L large compared with C, the support of F_j lies in the square of radius s'
about s' j. Let I_j be the open square of radius 4 s' about this center,
including the old auxiliary variables there. The first construction, the
REGULAR MAP, places no size restriction on the real f_i; its Taylor estimates
are used only near zero.
Fix D = 100. Join each pair of distinct parents at sup distance at most D by a
COMPARISON EDGE. Give each edge e = {j,k} a reference fair coin c_e, at its
midpoint, that favors one endpoint. A parent is SELECTED if all its incident
coins favor it. Write A_j for its selection indicator and
p = E_ref A_j = 2^{-((2D+1)^2 - 1)}.
Two selected parents have distance greater than D, hence their I_j are disjoint
with a free gap between them.
Order comparison edges first by length, then by independent continuous
priorities (the priorities belong to the environment). This rule is invariant in
law under translations and square symmetries. For an edge e, let d_e A_j be the
Doob martingale difference of A_j when c_e is revealed under the reference coin
law. It is zero unless e is incident to j and depends only on c_e and earlier
incident coins. If an earlier incident coin has already disfavored j, it is
identically zero for both outcomes of c_e. So its definition uses a finite local
star, regardless of the length of the global reveal order.
For fixed old field variables, give the new coins the successive conditional
likelihoods
( exp U_e ) / ( E_e exp U_e ), U_e = p^{-1} sum_j F_j d_e A_j, (7.1)
where E_e averages the current fair coin with all earlier coins fixed. Each
factor is positive and normalized, so the decoration preserves the original
marginal law of spins and old reference variables. The telescoping identity
sum_e d_e A_j = A_j - p gives the exact new logarithmic density:
sum_j F_j + sum_e ( U_e - log E_e e^{U_e} )
= p^{-1} sum_j A_j F_j - sum_e log E_e e^{U_e}. (7.2)
LEMMA 7.1 (Separation of the terms to be integrated).
Fix all new coins. Each normalizer log E_e e^{U_e} in (7.2) meets at most one
selected square I_j in its spin and old-auxiliary support. The same holds for
each selected term A_j F_j.
Proof. A parent whose F_i meets I_a is within 6 parent units of a. Suppose a
comparison edge has endpoint neighborhoods meeting two selected squares, centered
at a and b. An edge of length at most 12 cannot do this, since it would give
sup distance <= 24 < D. For a longer edge, an endpoint i within 6 of a selected
parent a != i has already lost the shorter comparison {i,a}, so its martingale
difference is identically zero. For both endpoint contributions to survive, both
endpoints would have to be selected, but their common comparison coin makes this
impossible. Therefore declare the support of a normalizer using only those
endpoint supports whose differences have not vanished. These declarations depend
on coins and priorities, not on the functions being Taylor-expanded. Selected F_j
themselves have radius less than one parent unit, so their separation is
immediate. QED.
Given the new coins, collect all terms in (7.2) that meet a selected square I_j
into H_j, and replace H_j by its conditional logarithm:
H_j -> log E_0[ e^{H_j} | field variables outside I_j ]. (7.3)
All other terms remain. The pure conditional laws in the selected squares are
independent given their exterior, so Lemma 7.1 shows, pointwise in the new
coins and exterior variables, that these replacements preserve the integrated
weight. The output still contains every spin, old auxiliary variable, and new
coin; making the updated old variables conditionally pure means their old law
is recovered by the conditional tilt e^{H_j}.
Allocate each term of (7.3) to its integration parent; allocate untreated terms
to their original parents, splitting a pair term equally between its endpoints.
These rules are fixed symmetrically once and for all. The allocated terms are
functions of the new coins and the remaining old field. Center each by its
full-plane pure/reference mean; the resulting potentials are f'_j. Centering
changes only a scalar partition factor (environment-dependent, field-
independent).
LEMMA 7.2 (Regular linearization).
The regular map is analytic on a fixed sup-norm neighborhood of zero once L is
fixed. Its derivatives of every prescribed finite order are bounded uniformly in
s and depend on a bounded local stencil of children. Its first derivative is
f'_{j,lin} = p^{-1} A_j sum_{i in C_j} P^0_{I_j} f_i, (7.4)
where P^0_{I_j} integrates the old reference field inside I_j. The transferred
sum is already centered, and (7.4) is independent of the reveal priorities.
Proof. Since E_e U_e = 0, every coin normalizer starts at degree two in the
child potentials. The derivative of (7.3) at zero is conditional expectation.
These facts give (7.4). Averaging the transfer in the full-plane reference law
gives zero by centering of each child, and averaging A_j preserves centering. On
a small ball, real conditional exponentials are uniformly bounded above and
below, so the ordinary power series for exp and log give the asserted sup-norm
bounds. There are only finitely many terms in each stencil, independently of s.
QED.
A moment estimate cannot assume every realized child potential is small, so
modify the map where child terms are large. This keeps exactness while making the
map's output controlled in high moments.
Flags. Fix b > 0 and independently attach to each child an environmental
threshold uniform on [b, 2b]. A child is a FLAG if its norm exceeds its
threshold. The continuous thresholds also make finite-step moment data
continuous in the initial coupling.
Inactive parents. Declare a parent INACTIVE when its center is within 10D parent
units of a flag. In (7.1) set its F_j to zero and retain its child terms
separately; its coins remain as dummy comparisons in the full local stars.
Active selected parents are integrated as before. Every retained term involving
an updated interior variable is included in that integration; retained terms
supported entirely outside may be left in place. Such a child term has support
much smaller than parent spacing and cannot meet two selected squares, so the
same exact identity holds at the boundary between active and inactive parents.
Isolated concentrations. Put d_0 = 4C + 10. An ISOLATED GROUP is a nonempty set
A of flags with child-grid diameter at most 6 d_0 and with no flag outside A
within 100D parent units of A. Distinct such groups are separated. Choose one
flag in A using independent environmental priorities, and let j(A) be its
parent. Let
G_A = sum_{i: dist(i,A) <= 3 d_0} f_i, I_A = I_{j(A)}. (7.5)
Taking L large puts the support of G_A deep inside I_A. All these children lie
in the inactive region, and no active integration, or coin normalizer involving
active old-field terms, reaches I_A.
G_A is removed while retaining all other interactions. Let B_A be the sum of the
remaining logarithmic terms that touch I_A. With the exterior fixed, replace G_A
by
R_A = log E_0[ e^{B_A + G_A} | I_A^c ] - log E_0[ e^{B_A} | I_A^c ], (7.6)
where I_A^c denotes all field variables not updated in I_A (new coins held
fixed). The identity
E_0[ e^{B_A + R_A} | I_A^c ] = E_0[ e^{B_A + G_A} | I_A^c ] (7.7)
shows the exterior weight is unchanged. The new conditional law inside I_A is
tilted by e^{B_A}, the old one by e^{B_A + G_A}; retaining B_A is part of the
exact replacement. Allocate R_A to j(A), and group all remaining terms at their
parents. These special integrations take priority over the rule of carrying
inactive terms; their separation makes their order immaterial.
PROPOSITION 7.3 (The local scale map).
There is a fixed support constant C such that, for all sufficiently large odd L
and every b > 0, the preceding construction maps a field with spacing s and
support/dependence radius Cs to a field with spacing s' = L s and
support/dependence radius C s'. Its outputs are centered, spin-flip even,
stationary on the parent grid, and square-symmetric in distribution. For a fixed
numerical K and a constant C_L,
||f'_j|| <= C_L sum_{i: ||s i - s' j|| <= K s'} ||f_i||. (7.8)
All replacements preserve exterior-integrated weights exactly, up to scalar
constants removed by centering, and admit the reversals described below. The
same formulas and centering constants apply in embedded plane and torus charts.
Proof. Choose C large enough to contain the comparison, inactivity, and
isolation ranges in parent units. Then choose L so large that old supports
overhang a parent by less than 1/10 parent unit and 100 d_0 / L < 1/10. Every
selection or status check, local allocation, and conditional partition function
then uses a fixed parent stencil. Old environment dependence adds only an
overhang C/L in these units, and the new priorities and thresholds are
independent and located in the same stencil. Increasing C once, before fixing L,
covers all supports; no integration requires environment variables beyond the
terms touching its finite square.
For any probability kernel P and real bounded H, |log P e^H| <= ||H||. This and
its Lipschitz version bound (7.3) and (7.6) by sums of input norms. The factor
p^{-1} and the finite numbers of terms depend only on the fixed step. Centering
at most doubles this bound, proving (7.8). Spin flip commutes with every
operation, and the symmetric choices and allocation rules give the stated
invariance in distribution. Exactness follows from the normalized decoration,
(7.3), and (7.7). Pure conditional kernels in an embedded square depend only on
its boundary assignment, so the same identities hold on a torus; always
subtract the same full-plane centering constants rather than torus means. QED.
COROLLARY 7.4 (Local implementation).
In a finite free chart W with arbitrary exterior field data, the map can be
performed away from its boundary so that the resulting specification agrees with
the full scale-s' specification in W eroded by K_1 s', for a fixed numerical K_1.
Proof. Use the full local comparison stars. If a parent's complete decoration
and allocation stencil leaves W, set its F_j to zero in every U_e and retain its
original child logs separately. Thus every fresh coin with a nontrivial
likelihood is inside W and is sampled. Skip integrations whose stencils leave W.
Include retained terms in any interior integration they meet. Skip special
integrations near the boundary, keeping active ones separated from them as above.
Every operation affecting the eroded chart then uses precisely its full stencil,
and all remaining discrepancies are supported in the removed boundary layer. The
argument also applies to annular charts. QED.
Two consecutive scales must be couplable so they differ only on small squares.
This is what turns exact marginals into geometric error estimates.
PROPOSITION 7.5 (A coupling of consecutive fields).
For each fixed environment, the input and output spin fields of the torus
construction, or of the local construction with prescribed old exterior field
variables, can be coupled so they agree outside squares of radius 5 s' about
marked parent centers. The marks are mutually independent conditional on the
environment, independent of the output Gibbs sample, and have local environmental
parameters p_j satisfying
p_j <= min{1, C_L sum_{i: ||s i - s' j|| <= K s'} ||f_i||}. (7.9)
For any P >= 1 and a stationary field with a finite Pth environmental moment,
E p_j <= C_L ( E ||f_0||^P )^{1/P}. These conclusions hold uniformly over the
old exterior assignments in the local implementation of Corollary 7.4.
Proof. First sample the entire output Gibbs field, including the new coins.
Independently sample a uniform variable at every candidate parent center, and
mark j when it is at most p_j = 1 - e^{-C_L S_j}, with S_j the environmental norm
sum in (7.9). These parameters depend only on the environment, so the marks have
exactly the independence stated, before any further conditioning.
Now condition on the new coins and the common exterior of the updated squares.
The ratio of the old conditional density to the new one is bounded below by
e^{-C_L S_j} with S_j the same local norm sum. This follows from the real log
bounds used in Proposition 7.3; for a special square it applies to the tilt from
B_A to B_A + G_A. If two laws satisfy mu >= (1-p) nu with p > 0, sample from nu,
retain that sample with probability 1-p, and otherwise sample from
[mu - (1-p) nu]/p. The Bernoulli decision can be independent of the nu sample;
when p = 0 the laws agree and the sample is retained. Use this residual
resampling in each marked updated square. Conditional factorization over the
separated squares proves the resampled field has the old conditional law. Unused
centers keep their dummy marks and cause no change. Each update lies inside its
radius-5s' square. Finally discard the conditionally decorated coins; their
decoration was normalized at each old field, so this recovers the input marginal.
Taking environmental expectation in (7.9) gives the last bound. QED.
COROLLARY 7.6 (Charge of a finite insertion).
For a finite family of labeled potentials at scale s define the normalized
insertion density and its charge by
D_f = exp( sum_i f_i ) / E_0 exp( sum_i f_i ), T_s(f) = q_s(D_f). (7.10)
The regular map satisfies T_{Ls}(f') = L^{-1} T_s(f). The identity holds
coefficientwise when separate indeterminates are attached to the input labels.
Proof. Only a finite neighborhood of the insertion has nonzero decorations or
replacement terms. Every observable outside a larger buffered region has the
same expectation against the normalized input and output densities, because the
decorations and replacements preserve its unnormalized pairing and the partition
function. Apply this to a pure bond tending to infinity. Theorem 6.4 identifies
the limiting pairing with Q, so the total lattice charge is unchanged. Dividing
by the output scale Ls instead of s gives the factor L^{-1}. Scalar centering
constants cancel in D_f. Analyticity near zero gives the coefficientwise
assertion. QED.
Next, estimate how the local map changes the size of the field in environmental
moments. The key distinction: the environmental MEAN has square symmetry and
gains the strong pure transfer exponent, while centered FLUCTUATIONS gain a
square-root saving from finite dependence. Rare large terms are handled by the
isolated integrations of Section 7.4.
Fix the even moment order P = 40. For a stationary scale-s field put
X = ( E ||f_0||^P )^{1/P}, Y = || E f_0 ||, m = E q_s(f_0),
W = sum_{i in Z^2} Cov( q_s(f_0), q_s(f_i) ). (8.1)
All norms are sup norms over field variables; expectations are environmental.
Finite dependence makes the covariance sum finite. Moreover W >= 0: if Lambda
is a growing square of cell indices then
(1/|Lambda|) Var( sum_{i in Lambda} q_s(f_i) ) -> W.
Primes denote the same coordinates for the output field at scale Ls; in
particular q_{Ls} = Q/(Ls).
PROPOSITION 8.1 (One-step estimates).
One can choose L large, then b > 0 small, and finally x_* > 0 such that for
every input field of Section 7 with X <= x_*,
|m' - L m| <= C_L X^2,
Y' <= rho Y + C_L ( |m| + X^2 ), (8.2)
X' <= rho X + C_L ( sqrt(W) + Y ),
where rho < 1 and all constants are independent of the scale s. The constants
C_L may depend on all fixed step parameters, including the subsequently chosen
b.
First establish the fluctuation estimate for the linearized map. The needed
fact is a dimension-free moment bound that permits taking a supremum over an
exponentially large spin boundary.
LEMMA 8.2 (Moments of finitely dependent Hilbert vectors).
Let (V_i), i in I with |I| = N, be centered random vectors in a real Hilbert
space. Suppose a bounded-degree graph on I is a dependency graph: subfamilies
with no graph edge between them are independent. Write
a = max_i ( E ||V_i||^P )^{1/P} and S = sum_i V_i. Then
E ||S||^P <= C_P ( E ||S||^2 )^{P/2} + C_{P,dep} N^{P/2-1} a^P. (8.3)
The constants do not depend on the dimension of the Hilbert space. In
particular ||S||_{L^P} <= C_{P,dep} N^{1/2} a.
Proof. Write P = 2r and expand ||S||^{2r} = ( sum_{i,j} <V_i,V_j> )^r. For each
tuple of 2r occurrences, connect two occurrences when their indices coincide or
are neighbors in the dependency graph. A singleton component has expectation
zero by centering and independence. If all components have size at least two and
one has size greater than two, there are at most r-1 components; bounded degree
gives at most C_{P,dep} N^{r-1} such tuples (choose one index per component, then
the rest along a spanning tree). Holder bounds each expectation by a^{2r}.
For the remaining tuples every component is a pair. Independence factors their
expectation into contractions of pair covariance operators. Sum these
contractions over all pairings and allow indices of different pairs to be
adjacent; the added tuples have merged components and incur the same
C_{P,dep} N^{r-1} a^{2r} error. The unrestricted sum is exactly the Wick
expression for E ||Z||^{2r}, where Z is a centered Gaussian Hilbert vector with
the covariance operator of S. That operator is positive even though individual
cross-covariances need not be. Diagonalizing gives
E ||Z||^{2r} <= C_P ( E ||Z||^2 )^r = C_P ( E ||S||^2 )^r. Finite-dimensional
projection and monotone convergence give the same in a general Hilbert space.
Finally bounded dependence gives E ||S||^2 <= C_dep N a^2. QED.
Fix a parent, translated to the origin, and transfer first to the boundary of
the square of radius 3Ls. Take the Hilbert norm there with respect to the pure
full-plane boundary marginal. The buffered density comparison (I1) bounds the
final conditional transfer from this boundary to dI_0 in sup norm by a fixed
constant times the Hilbert norm. Thus random Hilbert sums control the sup norm in
(7.4); the factor p^{-1} A_0 costs at most p^{-1}, fixed independently of L.
Let g_i = f_i - E f_i and t_i = q_s(g_i). Choose the bounded energy
representatives e_i = s P_{Cs,s i} B_{s i} (enlarging the fixed support
constant if needed; B_{s i} is the bond average of Theorem 6.4 at s i, and the
transfer is through a square centered there). Then Q(e_i) = s, q_s(e_i) = 1, and
||e_i|| <= C. Decompose
g_i = h_i + t_i e_i, Q(h_i) = 0.
Both h_i and t_i are centered in the environment, and
||h_i||_{L^P(E;inf)} + ||t_i||_{L^P(E)} <= C X.
The transferred h_i have sup norm at most C L^{-u} ||h_i|| by Theorem 6.4 (one
may first transfer across a square centered at the child and then condition onto
the intermediate parent boundary). There are L^2 children with a fixed
dependence range in child units, so Lemma 8.2 bounds their sum in L^P(E;L^2):
CL^{1-u} X. (8.4)
For the charge part, let w_i be the transfer of e_i to the same intermediate
boundary. These are deterministic Hilbert vectors with
||w_i|| <= C/L, ||w_i - w_j|| <= C L^{-u} if ||i-j|| <= 2C. (8.5)
For the second estimate, e_i - e_j has zero charge and is supported in a fixed
enlargement of one child neighborhood, so the zero-charge transfer estimate
applies.
The total covariance W gives a sharper bound than the variance of a single
child. Put c(h) = Cov(t_0, t_h) and I = C_0 (the children of parent 0).
Stationarity and finite dependence give
E || sum_{i in I} t_i w_i ||^2
= W sum_{i in I} ||w_i||^2
+ sum_{i in I, h: i+h in I} c(h) <w_i, w_{i+h} - w_i>
- sum_{i in I, h: i+h not in I} c(h) ||w_i||^2. (8.6)
Only a fixed number of h contribute, and |c(h)| <= C X^2. The first term is at
most C W because W >= 0 and sum_i ||w_i||^2 <= C. By (8.5) the second term is
O(L^{1-u} X^2); the last has only O(L) indices i near the block boundary, so is
O(L^{-1} X^2). Consequently
E || sum_{i in I} t_i w_i ||^2 <= C W + C ( L^{1-u} + L^{-1} ) X^2. (8.7)
Applying Lemma 8.2 with N = L^2 and a = C X / L gives
|| sum_{i in I} t_i w_i ||_{L^P(E;L^2)}
<= C sqrt(W) + C ( L^{(1-u)/2} + L^{-1/2} + L^{-2/P} ) X. (8.8)
This has no dependence on the spin boundary dimension. Together with (8.4) and
the final buffered transfer it implies
|| f'_{0,lin}(f - E f) ||_{L^P(E;inf)} <= eta_L X + C sqrt(W),
eta_L -> 0. (8.9)
All constants in the limit statement use only the support range and p, fixed
before L.
The exact normalization of Q makes the first line of (8.2) especially simple.
Applying q_{Ls} to (7.4) integrates the selection coins, so p^{-1} cancels
E_0 A_j = p. Charge is preserved by pure transfer. Therefore
E q_{Ls}(f'_{0,lin}) = (1/L) sum_{i in C_0} E q_s(f_i) = L m. (8.10)
For the environmental mean, let fbar_i = E_aux E f_i, where the auxiliary
expectation uses its independent reference law and leaves the spins fixed. Then
||fbar_i|| <= Y and q_s(fbar_i) = m. The deterministic spin function fbar_i is
quarter-turn invariant about its center, because both the environmental
distribution and the reference law respect this symmetry. The transfer inside
I_0 integrates all old auxiliary variables, so its mean is the transfer of
fbar_i. Subtract m e_i from fbar_i: the difference has zero charge and norm at
most C Y (since |m| <= C Y). Its transfer costs C L^{-w} Y per child, while the
charge part costs C_L |m|. Thus
|| E f'_{0,lin} || <= C L^{2-w} Y + C_L |m|. (8.11)
For the last line of (8.2), the simpler general transfer bound also gives
|| f'_{0,lin} ||_{L^P(E;inf)} <= eta_L X + C_L ( sqrt(W) + Y ). (8.12)
What follows records how flagged stencils affect low moments. Enlarge a parent stencil to
include every comparison, threshold, and isolation check that can affect its
output, and put S = sum_{i in Sc} ||f_i||. Its size depends only on the fixed
step. Let B be the event that it contains a flag. Markov and Holder give
Prob(B) <= C_{L,b} X^P, E[ S^r 1_B ] <= C_{L,b} X^P (1<=r<=P). (8.13)
For r < P the second follows from (E S^P)^{r/P} Prob(B)^{1-r/P}; for r = P it is
immediate. Choose b after L so small that 2|Sc| b lies in the uniform Taylor
neighborhood of Lemma 7.2. With no flag, every child norm is at most 2b, and the
regular Taylor remainder R satisfies
||R|| <= C_L S^2 <= C_L b S, (8.14)
the second with a constant independent of b. The raw bound (7.8) and (8.13) show
the actual map differs from its linearization by O_L(X^2) in first environmental
moment. Equations (8.10) and (8.11), with the charge bound of Theorem 6.4, now
prove the first two lines of (8.2) once L is large enough that C L^{2-w} < 1.
The same reasoning does not yet prove a contraction in the Pth moment: the
r = P bound in (8.13) has size X^P, not a higher power. The next argument uses
the LOCATION of flags to recover a small factor.
Consider one contributing child log f_i in the raw output stencil of (7.8).
Enlarge the determining stencil to contain the child-grid square of radius
100 D L + 2C about each such i, as well as every status check that can affect the
output. This enlargement contains the full isolation neighborhood of any flag
group lying within 2C of a contributing log. If a flag k in this stencil has
||i-k|| > 2C, then f_i is independent of that flag (fresh threshold included),
so
E[ ||f_i||^P 1_{k is a flag} ] = E||f_i||^P Prob(k is a flag)
<= C_b X^{2P}. (8.15)
A union bound over the fixed enlarged stencil makes all such contributions
O_{L,b}(X^2) in Pth-root norm.
It remains to consider contributions for which every flag in the stencil lies
within 2C of i. Those flags have diameter at most 4C. Including the full
isolation neighborhoods makes them an isolated group; all logs that can depend
on them lie within the 3 d_0 neighborhood transferred as G_A. Equivalently, if
isolation fails because of a distant extra flag, that flag is independent of the
contributing log and (8.15) applies. This dichotomy is the key; small flag
probability alone would not suffice.
For the isolated group, compare (7.6) with the same expression after the other
touching logs B_A are removed. For every probability kernel P_0,
| log( P_0 e^{B_A + G_A} / P_0 e^{B_A} ) - log P_0 e^{G_A} |
<= 2 ||B_A||. (8.16)
Each child log in B_A lies outside the group neighborhood and is independent of
an actual triggering flag. On the event that the group occurs, its Pth-root
contribution is therefore O_{L,b}(X^2) by (8.15); the same holds for every other
log omitted from the concentrated contribution.
After this removal, Lemma 6.6 gives
|| log E_0[e^{G_A}|I_A^c] - E_0 log E_0[e^{G_A}|I_A^c] ||
<= (C/L) sum_{i: dist(i,A) <= 3 d_0} ||f_i||. (8.17)
Indeed G_A lies in a patch of radius a fixed multiple of s. If the patch is not
concentric with I_A, transfer first to the square of radius 2Ls about the
representative flag (contained in I_A), then condition onward; averaging a
positive density cannot increase its log oscillation, and centering increases the
oscillation bound by at most a fixed factor.
There are a bounded number of flag patterns within a fixed child-radius
neighborhood of an anchor, and O(L^2) possible anchoring children for a given
output stencil. Separation bounds the number of simultaneous isolated groups in
that stencil by a constant independent of L. Raising (8.17) to the Pth power,
using the bounded number of logs per group, and summing over anchors gives
|| 1_B f'_0 ||_{L^P(E;inf)} <= C L^{2/P-1} X + C_{L,b} X^2. (8.18)
The first constant uses only the fixed child support radius, so is independent of
L and b. Do this without disjointness of the anchoring events: take their union
bound before the Pth root. The linear output on the same event obeys the same
bound, with a possibly different fixed constant: charge each child term to a flag
farther than 2C whenever one is present and use (8.15); otherwise only a bounded
child-radius cluster contributes a dependent large term, and the ordinary even
transfer to the parent boundary costs C/L for that cluster. The same O(L^2)
anchor count gives
|| 1_B (f'_0 - f'_{0,lin}) ||_{L^P(E;inf)}
<= C L^{2/P-1} X + C_{L,b} X^2. (8.19)
Fixed multiplicities from the allocation of terms are absorbed into constants.
Proof of Proposition 8.1 (completion). The last line follows by combining
(8.12), (8.14), and (8.19):
X' <= ( eta_L + C L^{2/P-1} + C_L b ) X + C_{L,b} X^2
+ C_L ( sqrt(W) + Y ).
Choose L large enough that the first two coefficients and the coefficient
C L^{2-w} in (8.11) leave a strict margin below one (possible since u>1, w>2,
P>2). Choose b next to put flag-free stencils in the Taylor neighborhood and make
C_L b smaller than the remaining margin. Finally choose x_* so that C_{L,b} X is
smaller than that margin for X <= x_*. Increasing the resulting contraction
factor to one common rho < 1 proves all three assertions. QED.
The mark parameters in Proposition 7.5 consequently satisfy E p_j <= C_L X.
One precise fact is recorded Taylor consequence needed later.
COROLLARY 8.3 (Fourth-order moment substitution).
Fix finitely many output stencils and a bounded multilinear expression in at most
four output potentials in total, allowing pure expectations, environmental
expectations, and products of such expectations. The multilinear kernels are
uniformly bounded in the scale; constants may depend on these kernels and their
fixed stencils. Substitute the regular map's Taylor polynomial through degree
four and retain only total input degree at most four. The resulting environmental
moment differs from that of the actual map by O_L(X^5), uniformly in s.
Proof. On unflagged stencils the fifth-order Taylor remainder and Holder give
O_L(X^5). On flagged stencils use the raw bound for the actual outputs and
(8.13). Products of four fourth-order Taylor polynomials have degree at most
16 < P, so the same bound controls every term there; terms of input degree 5 to
16 have expectation O_L(X^5) when X <= 1. Products of separately taken
expectations obey the identical estimates. QED.
Proposition 8.1 leaves one quantity uncontracted: the variance density W of the
energy charges. W is now replaced by a polynomial coordinate v_H whose leading
change under one scale step is a NEGATIVE multiple of W^2. The sign comes from
the normalization of a finite insertion: in a product of two independent centered
insertions, the normalizing denominator first affects the variance at degree four,
and its distant energy contribution is negative. Including the local cubic and
quartic insertion coefficients makes the calculation compatible with an exact
scale step.
Throughout, spacing is s and q_s = Q/s. All input potentials are centered under
E_0 (which includes auxiliary variables). Environmental expectation E does NOT
integrate these auxiliary variables; it does integrate the independent priorities
and thresholds used to build the scale map. Write R_0 for a fixed dependence
range in cell units, enlarged so disjoint cell sets at distance greater than R_0
use disjoint environment variables and have disjoint declared supports.
For a finite set of cell indices I and real indeterminates t = (t_i)_{i in I},
put
F_I(t) = sum_{i in I} t_i f_i,
T_I(t) = q_s( e^{F_I(t)} / E_0 e^{F_I(t)} ). (9.1)
For each fixed environment these are analytic near the origin, and make use of only
their Taylor coefficients, so no exponential integrability over the environment
is needed. Since E_0 F_I = 0 and q_s kills constants, the first three homogeneous
terms are
T_1(F) = q_s(F), T_2(F) = (1/2) q_s(F^2),
T_3(F) = (1/6) q_s(F^3) - (1/2) q_s(F) E_0 F^2. (9.2)
Hence the degree two, three, four terms of the environmental variance of T_I are
Var(T_1), 2 Cov(T_1,T_2), Var(T_2) + 2 Cov(T_1,T_3). (9.3)
All covariances are environmental unless an explicit subscript 0 appears.
A convention for coefficients is as follows, including repeated indices. For an
ordered tuple i = (i_1,...,i_d), give its d occurrences separate indeterminates
t_1,...,t_d, even if some i_a coincide, and define the symmetric multilinear
coefficient
C_d(i) = (partial_{t_1}...partial_{t_d}) at t=0 of
Var[ q_s( exp(sum_{a=1}^d t_a f_{i_a})
/ E_0 exp(sum_{a=1}^d t_a f_{i_a}) ) ],
2 <= d <= 4. (9.4)
The derivative means coefficient extraction using (9.2)-(9.3): a finite sum of
moments, valid even when the random potentials have only the finite moments of
Section 8.
DEFINITION 9.1 (Variance coordinate).
For H > R_0 define
v_H = sum_{d=2}^4 (1/d!) sum_{i_2,...,i_d in Z^2}
1_{ diam(0,i_2,...,i_d) <= H } C_d(0,i_2,...,i_d). (9.5)
The diameter is Euclidean in cell units. The coordinate is real and need not be
nonnegative for an arbitrary field. The factorial and ordered slots are the
usual Taylor convention; an equivalent definition sums over a large cell box and
divides by the number of cells, then passes to the limit. Stationarity and the
finite diameter cutoff make these agree. This latter definition also applies when
an intermediate allocation is periodic on parent cells: average over complete
parent periods before the limit.
LEMMA 9.2 (v_H is close to W).
For each fixed H > R_0 there is C_H < inf, independent of s, with
|v_H - W| <= C_H X^3 if X <= 1. (9.6)
Proof. The quadratic coefficient is C_2(i,j) = 2 Cov(q_s(f_i), q_s(f_j)), so its
contribution is exactly W (covariances vanish past the dependence range). Every
remaining coefficient has three or four potential occurrences whose supports fit
in a ball of radius O(Hs). The charge bound (6.10), boundedness of pure
expectation, and Holder bound each degree-d coefficient by C_H X^d. There are
finitely many tuples with first slot zero and diameter at most H. QED.
First change H to LH WITHOUT changing the field. This calculation contains the
sign of the flow. Later this larger cutoff is compared with the coordinate of the
output field.
LEMMA 9.3 (Shell calculation).
Fix finite A, B. If 0 < z <= 1, X <= A z, Y <= B z^2, then
v_{LH} - v_H
= -2 kappa W^2 sum_{y in Z^2: H < |y| <= LH} |y|^{-2}
+ e_H z^4 + O_H(z^5), (9.7)
where |e_H| is bounded by a deterministic function tending to zero as H -> inf.
The constants may depend on A, B and the fixed step parameters, but are uniform
in s.
Proof. Decompose f_i = fbar_i + ftilde_i with fbar_i = E f_i. An occurrence of
ftilde_i has size order one; an occurrence of fbar_i has size order two.
Products of total size order at least five contribute O_H(z^5) by Holder. For the
remaining coefficients connect two occurrences when their cell indices are within
R_0. A connected component with at most four occurrences has bounded diameter,
and a single centered occurrence has zero environmental expectation in every term
of the covariance expansion. So for H large a nonzero coefficient in the shell
has exactly two distant groups: either a centered pair and one mean, or two
centered pairs, each of total size order four. This also applies to the products
of means subtracted in a covariance.
Call the two groups A and B. Let 1 + g_A and 1 + g_B be their separately
normalized insertion densities, expanded formally in their own indeterminates;
every coefficient of g_A, g_B is pure centered. Define
T_A = q_s(g_A), T_B = q_s(g_B), h_{AB} = q_s(g_A g_B),
c_{AB} = E_0(g_A g_B). (9.8)
Multiplying the two densities and normalizing again gives the exact formal
identity
T_{A union B} = ( T_A + T_B + h_{AB} ) / ( 1 + c_{AB} ). (9.9)
If the anchor distance is l, Proposition 6.5 gives, coefficient by coefficient,
h_{AB} = O(l^{-u}), c_{AB} = l^{-2} ( kappa T_A T_B + o(1) ). (9.10)
The estimates carry the product of the norms of the local coefficient functions;
all group diameters are bounded, so the constants are uniform in the slots and in
s.
The variance of T_A + T_B has no coefficient using both independent groups.
Contributions with two cross factors h_{AB} or c_{AB} have shell sum bounded by
C H^{2-2u} z^4 = o_H(1) z^4. For a single h_{AB} make use of the stronger symmetric
part of Proposition 6.5. To justify it, replace the cutoff on total diameter by
the cutoff on anchor distance and sum the bounded relative positions inside each
group; environmental factorization turns the coefficient into a finite sum of
bilinear charge pairings of deterministic functions. Besides fbar_x, the two
possible local sums are
D_x = sum_{|a| <= R_0} E[ q_s(ftilde_x) ftilde_{x+a} ],
M_x = sum_{|a| <= R_0} E[ ftilde_x ftilde_{x+a}
- E_0(ftilde_x ftilde_{x+a}) ]. (9.11)
The pair-plus-mean sector gives pairings q_s(D_x fbar_y); the pair-plus-pair
sector gives q_s(D_x M_y), with their exchanges and the fixed scalar factors
from Taylor expansion. This list comes from taking the linear charge from one
group and the term linear in that group in h_{AB}; the other group supplies its
mean or its quadratic normalized density. Terms leaving a centered singleton
vanish as above. All displayed functions are pure centered, and D_x, M_x have
norm O(z^2). Rotating the offset sum preserves its index set and the environment
law, while charge is a rotation-invariant scalar, so each function is
quarter-turn invariant about its anchor. The resulting bound is C l^{-w} z^4,
whose shell sum is O(H^{2-w}) z^4 = o_H(1) z^4 (w > 2).
Replacing diameter by anchor distance affects only strips of bounded thickness at
the two shell ends. There are O(H) possible anchor displacements there, and the
unsymmetrized bound O(H^{-u}) gives an error O(H^{1-u}) z^4. So replacement and
reversal both contribute vanishing error as H -> inf.
Only the single c_{AB} contribution remains. Up to size order four it is
-2 Cov( T_A + T_B, (T_A + T_B) c_{AB} ). (9.12)
Its four occurrences must all be centered (a mean would raise the size order).
Write U = T_{A,1} and V = T_{B,1}, independent centered random scalars. At
bidegree (2,2),
-2 Cov( U + V, (U + V) kappa l^{-2} UV )
= -2 kappa l^{-2} E[ (U+V)^2 UV ]
= -4 kappa l^{-2} E U^2 E V^2. (9.13)
The o(1) in (9.10) gives o_H(1) z^4 after shell summation, since L is fixed and
sum_{H<|y|<=LH} |y|^{-2} is bounded.
The slot count is as follows. Put a_{ij} = Cov(q_s(f_i), q_s(f_j)). Suppose the
four labeled slots split into the far pairs {1,2} and {3,4}. The coefficient of
t_1 t_2 in E U^2 is 2 a_{i_1 i_2}, even when i_1 = i_2 (the occurrences still
have distinct indeterminates); the other pair contributes 2 a_{i_3 i_4}. Hence
the polarized normalization contribution is
-16 kappa |i_3 - i_1|^{-2} a_{i_1 i_2} a_{i_3 i_4}.
There are three partitions of four labeled slots into two unordered pairs; their
sums agree by permuting dummy slots, and a tuple in a sufficiently distant shell
has at most one such far pairing. The prefactor from (9.5) is 1/4!, so the
numerical coefficient is -16 * 3 / 4! = -2. For the displayed pairing anchoring
i_1 = 0 and writing y = i_3 gives
sum_{i_2} a_{0 i_2} sum_{i_4} a_{y i_4} = W^2,
since both sums are finite and stationarity makes the second equal to W. Changes
of the cutoff from bounded shifts of anchors have the strip bounds already proved.
A change of anchor in the kernel itself also vanishes: for bounded delta and
|y| comparable to H, | |y+delta|^{-2} - |y|^{-2} | <= C|delta| H^{-3}, whose
shell sum is O(H^{-1}). This proves (9.7). QED.
Let f'_j be the output potentials at spacing Ls, and let v'_H be their
coordinate. A finite insertion has the same exterior marginal before and after
the regular scale map, and this is why the truncated, locally allocated
coordinate behaves.
LEMMA 9.4 (Compatibility of the coordinate).
Under the hypotheses of Lemma 9.3,
v'_H - v_{LH} = e_H z^4 + O_H(z^5), (9.14)
with the same uniformity convention for e_H.
Proof. First use the regular map and retain the child indeterminates. For finitely
many labeled inputs its normalized total insertion charge obeys
T_out = L^{-1} T_in (9.15)
by Corollary 7.6. This can also be seen directly: the conditional coin likelihood
integrates to one at every fixed old configuration, and each conditional
integration preserves pairing with a test outside the affected region; pairing
with a distant pure bond identifies Q by Theorem 6.4. Centering the allocated
output logs changes only a scalar, which disappears from the normalized insertion;
the factor L^{-1} is exactly the change from Q/s to Q/(Ls).
This holds for every fixed realization of the fresh priorities. Taking
environmental variance and then Taylor coefficients therefore preserves complete
coefficients, including their mean subtractions. The variance factor L^{-2}
cancels the L^2 children per parent when sums are taken per unit area; if an
allocation distinguishes positions within a parent, take the area average over
complete parent periods. No invariance of individual allocated coefficients under
child translations is assumed.
The two cutoffs are compared. Every output Taylor coefficient uses children in a
fixed stencil, so classifying a coefficient by parent diameter H rather than child
diameter LH changes its classification only when the child diameter lies in a
strip of width O_L(1) about LH. Keep together the full covariance coefficient for
each fixed tuple of output arguments (all mean subtractions included), and only
then expand its arguments in child slots; the parent cutoff is constant on
precisely this grouping. Write R_j for the regular output at parent j as a
function of its child potentials. A Taylor contribution with m output arguments
and child degree d, 2 <= m <= d <= 4, assigns its d occurrence slots to nonempty
blocks B_1,...,B_m and uses the random functions
G_a = D^{|B_a|} R_{j_a}(0)[ f_{i_b}: b in B_a ], 1 <= a <= m. (9.16)
The directions are supported at their indicated child labels (repeated labels
allowed). Multilinear substitution gives the complete polarized covariance
coefficient of G_1,...,G_m, with its Taylor factors and the single common weight
1_{diam(j_1,...,j_m) <= H}.
Discard size orders at least five at cost O_H(z^5), and connect child slots when
their enlarged stencils can share environment variables. A singleton centered
child has zero expectation even conditional on the fresh priorities: the Taylor
kernel depends on those independent priorities and takes the child function as a
multilinear argument. A singleton mean costs size order two. Two components
containing only means cannot create a lower-order term through random allocation;
at the only relevant order their output is the linear map of Lemma 7.2, which is
independent of priorities, with selection indicators integrated by Q (so their
charges are deterministic when inputs are). Remaining are two distant
components of bounded diameter, size order two each.
Every output argument receiving a child slot lies in that slot's bounded stencil,
so the components partition the relevant output arguments into two distant groups
as well: each block B_a of (9.16) lies wholly in one component. Apply (9.9) to the
two finite lists of output arguments G_a with separate indeterminates u_a. The
groups are independent in the environment, including their fresh priorities. For
their separate charges,
Var( T_A(u_A) + T_B(u_B) ) = Var T_A(u_A) + Var T_B(u_B)
has zero coefficient involving variables from both groups, regardless of their
separate means. So the unmixed contribution cancels for each fixed output tuple
and each child-block partition, before any sum over output labels. Repeated child
labels merely repeat derivative directions inside a group; no within-group
independence is used. The remaining terms have a cross factor and so an O_L(H^{-u})
bound from Proposition 6.5. The strip therefore costs O_L(H^{1-u}) z^4 =
o_H(1) z^4. Keeping the covariance subtractions with each coefficient is essential
and permitted because the cutoff depends only on the tuple of labels.
Finally apply Corollary 8.3. For each fixed output tuple, (9.3) is a sum of
bounded multilinear expressions in at most four output potentials, including
products of separately taken expectations. The charge bound on its fixed-H support
makes the kernels uniformly bounded in s. The corollary applies to the finite
union of all determining stencils, including status checks, so it replaces the
regular Taylor output by the actual output and removes child degrees above four
with total error O_H(X^5) = O_H(z^5). The highest intermediate degree is
4*4 = 16 < P = 40, so the finite moment assumption suffices. This proves (9.14).
QED.
PROPOSITION 9.5 (Marginal variance flow).
Fix the scale-step parameters and finite constants A, B. For every sufficiently
large fixed H, the polynomial coordinate of Definition 9.1 satisfies, whenever
0 < z <= 1, X <= A z, Y <= B z^2,
v'_H - v_H
= -2 kappa W^2 sum_{y in Z^2: H < |y| <= LH} |y|^{-2}
+ e_H z^4 + O_H(z^5), (9.17)
where |e_H| -> 0 as H -> inf, uniformly in the lattice scale. Moreover
|v_H - W| <= C_H X^3 for X <= 1.
Proof. Combine Lemmas 9.2, 9.3, and 9.4. QED.
The shell sum tends to 2 pi log L as H -> inf, so the leading coefficient in
(9.17) is strictly negative. In the next section the temperature is chosen so that
the mean charge stays small enough for this variance decay to continue at every
scale.
A temperature is chosen parameter for which the scale iteration stays small and
its potentials tend to zero in environmental moments. This is an EXISTENCE
argument for the iteration; its identification with the physical critical
temperature comes in Section 12.
Write the dimensionless nearest-neighbor coupling as
K(t) = (1/2) log( 1 + sqrt2 e^t ), K_e = K(h + theta xi_e),
theta > 0. (10.1)
Then K(0) = (1/2) log(1 + sqrt2) = K0, and
log( (e^{2K_e} - 1)/sqrt2 ) = h + theta xi_e. The signs xi_e are the independent
symmetric ones of the model. This coordinate makes duality especially simple.
Fix an initial spacing s_0, large enough for the pure estimates and the scale
construction. Allocate each bond to its nearest cell center, splitting ties
equally. If alpha_{i,e} is the resulting nonnegative allocation, then
sum_i alpha_{i,e} = 1, and the rule is equivariant under grid translations and
square symmetries. Set
f_i^(0) = sum_e alpha_{i,e} ( K_e - K_0 ) b_e,
b_e = sigma_x sigma_y - E_0( sigma_x sigma_y ). (10.2)
These potentials are pure centered, and their sum is precisely the change in the
Ising log weight, modulo a spin-independent constant.
LEMMA 10.1 (Initial moment bounds).
Let a = K'(0) > 0. For every fixed T < inf, uniformly for |h| <= T theta^2, as
theta -> 0,
W_0 = 2 a^2 theta^2 + O(theta^3), X_0 = O_{s_0}(theta),
Y_0 = O_{s_0}(theta^2 + |h|), partial_h m_0(0,0) = 2 s_0 a > 0. (10.3)
For every fixed cutoff H > R_0, v_{H,0} = 2 a^2 theta^2 + O_H(theta^3) on such an
interval.
Proof. There are two bonds per lattice site, so periodicity of the allocation
gives sum_e alpha_{0,e} = 2 s_0^2. Since Q(b_e) = 1,
m_0 = 2 s_0 E[ K(h + theta xi) - K_0 ].
Independence of distinct bond signs and sum_i alpha_{i,e} = 1 give the exact
identity
W_0 = s_0^{-2} sum_e alpha_{0,e} ( sum_i alpha_{i,e} )
Var( K(h + theta xi) ) = 2 Var( K(h + theta xi) ).
Taylor expansion of the smooth function K proves the claimed asymptotics and
derivative. The finite number of bounded bond functions in a cell gives the X_0,
Y_0 bounds, and Lemma 9.2 gives the v_{H,0} assertion. QED.
The charge mean has an expanding linear term L m, while the variance has a
negative quadratic correction (Prop 9.5). The initial h is chosen so that the
expanding coordinate never leaves a narrow interval of width proportional to the
running variance.
THEOREM 10.2 (A small critical trajectory).
Fix the scale-step parameters chosen in Proposition 8.1 and a sufficiently large
initial spacing s_0. There are constants A, B, D, c > 0, a fixed cutoff H, and
theta_0 > 0 such that, for each 0 < theta < theta_0, one can choose a
deterministic h = h(theta) = O(theta^2) with the following property. The iterates
at spacings s_k = s_0 L^k satisfy, with v_k = v_{H,k},
v_k > 0, X_k <= A sqrt(v_k), Y_k <= B v_k, |m_k| <= D v_k, (10.4)
and
v_k <= 1 / ( v_0^{-1} + c k ). (10.5)
In particular X_k, Y_k, m_k -> 0, and their uniform bounds over all k can be made
arbitrarily small by decreasing theta_0.
Proof. Choose the constants so the moment bounds propagate whenever the mean stays
between the walls m = +/- D v. In Proposition 8.1 let C_1 be a common constant
and keep its contraction rho < 1. Choose A large enough for the initial
X_0/sqrt(v_0) bound and so that
rho A + 2 C_1 < A. (10.6)
Choose D next with
(L - 1) D > C_1 A^2 + 1. (10.7)
Choose T next so that, for every subsequently fixed H, the initial ratios
m_0/(D v_{H,0}) at h = -T theta^2 and h = T theta^2 are strictly below -1 and
above 1 for small theta. Lemma 10.1 permits this independently of H: the leading
denominator is 2 D a^2 theta^2, while the leading numerator divided by theta^2 is
affine in h/theta^2 with positive slope 2 s_0 a. Finally choose B large enough
that
rho B + C_1 (D + A^2) < B (10.8)
and Y_0 <= B v_{H,0} for |h| <= T theta^2 and small theta, for every subsequently
fixed H (possible from Y_0 <= C_{s_0}(1+T) theta^2 and v_{H,0}/theta^2 -> 2a^2).
So far the order is A, D, T, B.
For data satisfying (10.4), Lemma 9.2 gives
W = v + O_H( v^{3/2} ), (10.9)
with the implicit constant including A^3. Use Proposition 9.5 with z = sqrt(v);
its constants A, B are now fixed. Choose H large enough that the e_H v^2 error is
smaller than one quarter of the negative leading term (using the positive limit
2 pi log L of the shell sum). Then restrict v to a small interval, also within the
smallness of Proposition 8.1, so that (10.9) and the O_H(v^{5/2}) remainder yield
-C_2 v^2 <= v' - v <= -c v^2, v' > 0, v'/v = 1 + O(v), (10.10)
with c, C_2 > 0 fixed and the smallness interval uniform in scale. Positivity
follows, e.g. from C_2 v < 1/2.
The X estimate now gives
X' <= rho A sqrt(v) + C_1 sqrt(W) + C_1 B v
<= ( rho A + C_1 + o(1) ) sqrt(v) <= A sqrt(v'),
using (10.6) and v'/v -> 1. Similarly Y' <= (rho B + C_1(D + A^2)) v <= B v' by
(10.8). At the upper wall the mean estimate gives m' >= L D v - C_1 A^2 v > D v';
at the lower wall m' < -D v'. The smallness interval was reduced to use (10.7) and
(10.10). Thus the walls point outward while both norm bounds propagate up to and
including the first exit step.
Now restrict theta so the initial interval [-T theta^2, T theta^2] has all the
properties fixed above and lies in the uniform smallness range just obtained. On
it v_0 = 2 a^2 theta^2 + o(theta^2) uniformly. The initial X bound is covered by A,
because changing h by O(theta^2) does not change the leading O(theta) bound on
X_0. The complete parameter order is A, D, T, B, H, then the small interval for
theta.
All moment data are continuous in h at every finite number of steps. To see the
only possible discontinuities, fix h and couple the constructions for nearby
parameters using the same environment, priorities, and threshold variables. At a
threshold comparison the threshold is independent of the compared data and has a
continuous distribution, so equality has probability zero. At finitely many steps
only finitely many comparisons enter any fixed local stencil, so the potentials
are a.s. continuous at the fixed h. The crude real-log bounds of Proposition 7.3
dominate them on every compact initial-parameter interval by a finite deterministic
bound at each fixed number of steps, so dominated convergence applies to X, Y, m,
W, and the finite polynomial v_H.
Continuity first supplies a closed initial subinterval I_0 on which
-1 <= m_0/(D v_0) <= 1, with endpoints mapped to -1 and 1. Suppose I_k has been
chosen so all parameters in it satisfy (10.4) through time k, with endpoints on
opposite time-k walls. The estimates just proved give positive v_{k+1} and the two
norm bounds throughout I_k, and its endpoints map strictly outside the opposite
time-(k+1) walls. The intermediate value theorem gives a closed subinterval
I_{k+1} subset I_k on which -1 <= m_{k+1}/(D v_{k+1}) <= 1, endpoints on the two
walls (take the last crossing of the lower wall before a first subsequent crossing
of the upper wall). The preceding-time bounds remain valid on this smaller
interval. Compactness gives a parameter h in the intersection of all I_k; no
monotonicity of an iterated coordinate in h is needed.
For this parameter (10.10) holds at every step. Taking reciprocals gives
1/v_{k+1} >= 1/v_k + c, which proves (10.5). The other conclusions follow from
(10.4) and v_0 = O(theta^2). QED.
Every constant in Theorem 10.2 is chosen before a domain or a mesh approximation
is specified. The resulting decay need NOT be summable over scales. Section 11
uses the locality of each scale step to attach a geometric weight to its
contribution to a fixed macroscopic crossing test, which is what makes the
non-summable errors harmless.
The trajectory of Theorem 10.2 has X_k -> 0, but its decay need NOT be
summable. It is proved that every fixed finite collection of macroscopic crossing and
passage tests nevertheless has asymptotically the same law as in the pure model.
The gain: a change confined to a small rescaled square can alter a macroscopic
test only when several long contour passages reach far out of that square, and the
clean model bounds each such passage by a positive power of the size ratio.
Summing these geometrically weighted errors makes the total comparison tend to
zero. An exterior supremum is taken first, so distinct ring neighborhoods multiply.
Use the triangulation obtained by drawing the northwest-southeast diagonal in
every lattice square, and interpolate the spins affinely. Its zero contours
implement the prescribed north-east / south-west resolution up to a matched
displacement of order one lattice spacing. A COLORED PATH uses vertices of one
spin sign and edges of this triangulation. A POLYGONAL CORRIDOR is a closed
topological rectangle with two opposite boundary arcs designated as gates; its
crossing test asks for a colored path between the gates. A POLYGONAL BAND is the
region between two disjoint simple polygons; its circuit test asks for a colored
path separating its boundary components. These sets are approximated by subcomplexes
of the triangulation, preserving the cyclic order of their marked sides, with
error of order one lattice spacing. The polygons are fixed and nondegenerate, and
every test has a specified open neighborhood in which spins remain free.
A CONTOUR PASSAGE between two terminal sets is a contour subarc joining them whose
interior lies in the prescribed permitted region. Different passages have disjoint
open parameter intervals on each contour; their endpoints may coincide at terminal
hits, so repeated passages of one contour are counted separately. Cropping at the
last visit to the first terminal before the first visit to the second gives this
convention (the passage convention of [S, Sec 5]).
For a center z, direction alpha, and 0 < r < R, put
S_r(z,alpha) = z + e^{i alpha} [-r,r]^2,
A(z,alpha;r,R) = S_R(z,alpha) \ interior S_r(z,alpha).
Make use of three checks:
F_4(z,alpha;r,R) = { at least four passages across A(z,alpha;r,R) },
H_2(z,alpha;r,R) = { at least two such passages confined to
z + e^{i alpha}{ Im w >= 0 } }, (11.1)
F_1(z,alpha;r,R) = { at least one passage across A(z,alpha;r,R) }.
The straight line in H_2 is a confinement wall inside a bulk chart; no spins are
pinned on that line. Its terminals are the inner and outer square boundaries.
Fixed proportional crops, and fixed unions covering the two possible half-planes,
are included whenever the geometry requires them.
Deterministic clean inputs (from [S]). For pure critical spins in
buffered bulk charts,
limsup_{delta->0} Prob_0(F_4(r,R)) <= C (r/R)^{2+c_0},
limsup_{delta->0} Prob_0(F_1(r,R)) <= C (r/R)^{c_0}, (11.2)
limsup_{delta->0} Prob_0(H_2(r,R)) = o(r/R),
where c_0 > 0 and the ratio is fixed before the mesh limit. These are
[S, Sec 2, "Iterated four-passage bound" and "Pure buffered RSW consequences";
Sec 5, "Two bulk passages confined to a half-plane"]. The conditional density
comparison [S, Sec 2, "Two collars around an annulus"] bounds the law on an
annular support under arbitrary pins outside fixed proportional inner and outer
collars by D times its pure bulk marginal, with D not growing with the annulus
ratio. These inputs also allow translations, rotations, and the fixed crops above.
Choose a fixed gap factor G large enough to separate the collars in successive
rings. Fix in advance smaller proportional collars that will survive the boundary
erosions. Include in D the comparison constant for those collars and the finite
losses for crops and half-plane choices, then choose a large fixed ratio a. The
bounds (11.2) give exponents and positive tolerances satisfying
p_4 > 2, p_h > 1, p_1 > 0, q_i^0 + tau_i < (Ga)^{-p_i}
for i = 4, h, 1, (11.3)
where q_i^0 already includes the conditional density loss. For the middle
inequality, first use o(a^{-1}) to get strict slack below (Ga)^{-1}, then choose
p_h > 1 close enough to one; no specified half-plane power law is required.
Increase the starting scale so the same q_i^0 bounds hold at finite mesh for all
three fixed shapes of ratio a. Uniformity in the frame follows by compactness of
directions and normalized offsets, using slightly relaxed radii. All these choices
precede the choice of the weak-randomness interval.
The clean geometric implication [S, Sec 5, "Passages forced by a
pivotal square"]. For a corridor crossing, a band circuit, or one of the three
checks T, there is a positive geometry scale a_T such that a change confined to a
square of radius b < a_T about z can change T only if its common exterior contours
supply four passages between scales b and u, two half-plane passages between u and
v, and one passage between v and a_T, where
u = min{ a_T, max{ b, dist(z,E_T) } },
v = min{ a_T, max{ b, dist(z,V_T) } }. (11.4)
E_T and V_T denote the unions of the sides and vertices of its polygons, so
b <= u <= v <= a_T. The scale a_T is smaller than every relevant side length,
terminal separation, and separation of nonincident sides. A fixed number of rings
may be omitted near transitions; half-plane rings may be recentered on a side and
the last rings near a vertex. For each of the three check shapes, a_T is a fixed
small multiple of its inner radius r; choose it small enough that every retained
ring has inner radius at most r/2 and its collars remain in the prescribed free
neighborhood. This implication concerns arbitrary fillings of the changed square,
so it also applies to a square whose values are chosen after other edits.
[Figure: three ranges of checks forced by a pivotal square. Blue rings have
separated dotted collars; their centers may change between ranges. Orange denotes
the edit square, a side, and a vertex of the test. Enlarged environment
neighborhoods fit in these separated collars, and the first retained ring avoids
the environment stencil of the inserted mark.]
In this subsection lengths are lattice lengths unless a factor delta is displayed.
Write eta for the full environment, including the independent priorities and
thresholds used to construct the potentials. A Gibbs variable includes both spins
and reference auxiliary variables, denoted chi. At level k, in a finite free chart
W with exterior configuration zeta, the specification is
mu_{k,eta,W}^zeta(d chi_W) = exp( F_{k,W}^zeta(chi_W) )
/ mu_{0,W}^zeta( exp F_{k,W}^zeta ) * mu_{0,W}^zeta(d chi_W),
F_{k,W}^zeta = sum_{j: supp f_j^(k) intersects W} f_j^(k)(chi_W zeta_{W^c}). (11.5)
The reference specification is pure critical Ising times its independent reference
auxiliary variables. Conditioning again is possible in a smaller free chart. All suprema
below include every exterior spin and auxiliary assignment, so they also bound
conditional laws with a random exterior.
Corollary 7.4 permits run the local construction inside a free chart. The
prescribed exterior zeta contains only level-k field variables. If a parent's
complete decoration and allocation stencil leaves W, set its F_j to zero in every
new coin likelihood and retain its child logs; keep the full comparison stars and
the dummy coins. Every fresh coin with a nontrivial decoration is then sampled
inside W. Integrate only when the full stencils lie in W. Frozen terms, including
terms entering from outside W, are included in any integration they meet; the
exclusion rule still prevents a term from meeting two integration squares. Skip
special integrations near the boundary by the same rule. The resulting identity is
exact, and its conditional specification agrees with the full level-(k+1)
specification at distance at least C_1 s_{k+1} from the boundary. Only after
constructing this joint output law does conditioning outside the eroded interior,
whose exterior may now include fresh coins. Disintegration gives a mixture of
level-(k+1) conditional specifications, each bounded by the corresponding exterior
supremum. This uses local conditional identities and never conditions on a
favorable boundary event.
For a check on A(z,alpha;r,a r) take the surrounding chart
W = S_{2 a r}(z,alpha) \ S_{r/2}(z,alpha). To iterate in this fixed chart, let the
available chart at level k be W_k = W minus M_1 s_k (removing points within M_1 s_k
of the boundary). Choose M_1 so M_1 s_{k+1} >= M_1 s_k + C_1 s_{k+1}; it may also
absorb allocated-support overhangs. Choose M_0 much larger than M_1 and all
stencil constants. Then a check with inner radius r >= M_0 s_k retains the smaller
proportional collars chosen in 11.1, inside W_k, and successive interior steps
retain them while the check is still in this range. Raise the first retained radius
if needed so the fixed ring gaps separate all environment stencils as well as the
free charts. The factor G and the slack in (11.3) remain fixed.
By Proposition 7.5, reversing step k can change spins only in squares of radius
b_k = 5 s_{k+1}, E p_{x,k}(eta) <= C_L X_k , (11.6)
centered on the parent grid (the factor five absorbs rounding). Conditional on
eta, these Bernoulli marks are mutually independent and independent of the entire
new Gibbs sample, including its fresh coins, before any further conditioning.
Their parameters depend only on environment variables within a fixed multiple of
s_{k+1} of the center. They can be placed at every candidate center, independently
of which integrations were selected. The same upper parameter envelope works for
the localized construction.
For clarity, the elementary reason for this domination is a lower likelihood-ratio
bound. If S_x is a fixed constant times the sum of child norms in the update
stencil, then, conditional on the fresh coins and the common exterior of the
integration squares, the before-integration law has density at least e^{-S_x}
relative to the after-integration law. It is therefore a mixture, with weight
e^{-S_x}, of the new law and a residual probability law. On a Bernoulli mark of
parameter 1 - e^{-S_x} use the residual law; otherwise retain the new sample.
Uniformly in all boundary data, 1 - e^{-S_x} <= min{1, S_x}, giving (11.6).
Conditional separation of the update squares permits independent marks, including
for the special ratio integrations. Integrating the discarded fresh coins out of
the reconstructed decorated law recovers the old spin law with its originally
prescribed level-k exterior.
Given a spin configuration sigma and marks M from just one reversal, let
C(sigma,M) be the configurations agreeing with sigma off the marked squares. For a
Boolean test T put
T^+(sigma,M) = 1{ T(sigma~) = 1 for some sigma~ in C(sigma,M) },
D_T(sigma,M) = 1{ T is not constant on C(sigma,M) }. (11.7)
Both are increasing in the marks, irrespective of spin monotonicity. For a check T
of inner radius r in its annular chart, define
B_{k,T}(eta) = sup_zeta mu_{k,eta,W_k}^zeta( T ),
B^+_{k,T}(eta) = sup_zeta
( mu_{k,eta,W_k}^zeta tensor pi_{k-1,eta} )( T^+ ), (11.8)
where pi_{k-1,eta} is the independent mark law for reversal of step k-1. At k = 0
only the plain bound is used. The supremum is taken before environmental
expectation. Only finitely many exterior field variables enter these finite
charts, and the auxiliary variables have finite alphabets, so these suprema are
measurable finite maxima. Each quantity is a function only of the environment in
the chart and a fixed O(s_k) neighborhood: the supports of the potentials touching
the chart and of the mark parameters whose squares meet the test. Mixtures over
exterior data need not have this locality; the suprema do, and this is exactly
why make use of suprema.
The next lemma supplies the single-step error. Its hypothesis is proved by
induction in 11.4, first for the three fixed check shapes, then used for any
prescribed polygonal test.
LEMMA 11.1 (A single-level pivotal bound).
Let spins have the level-k specification in a free chart, and allow marks from
step k-1 of radius b = 5 s_k. Suppose the possibility envelope for every retained
ring check satisfies E B^+_{k,T_i} <= q_i^0 + tau_i. The rings start at a
sufficiently large fixed multiple of b and have disjoint collars enlarged by all
spin, mark, and environment stencils. Then the expected conditional supremum of
the probability that a fixed test T is changed by these marks is at most
C_T X_{k-1} min{ 1, (s_k / a_T)^c }, (11.9)
for a fixed c > 0. The same bound controls the increase from T to T^+. Constants
are uniform over translates, rotations, and dilates of each of the three check
shapes. Any deterministic partial activation of the marks satisfies the same
estimates.
Proof. Activate the finitely many relevant marks in deterministic order. For
fixed eta and exterior data, conditioning on all previous marks charges the first
instability at x to p_{x,k-1}(eta) times a pivotal possibility event with that
mark omitted. If adding its square first permits two test values, choose a
witnessing assignment and revert the new square, retaining all other assigned
values; the resulting pair differs only in that square. Every passage required
outside it by the geometric implication (11.4) is consequently possible using only
the previous marks, and these may be enlarged to all other marks in their
respective ring neighborhoods.
Consider a retained family of rings with checks T_1,...,T_m. For fixed environment,
condition off the first ring's eroded chart W_{1,k} and on the marks whose squares
do not meet its spin support. The other ring checks are measurable under this
conditioning; the remaining spins have the specified conditional Gibbs law, and
the remaining marks retain their independent law. Its factor is at most
B^+_{k,T_1}(eta). Remove each ring in turn. Pointwise in the original exterior
data this proves
sup_zeta [ p_{x,k-1}(eta) Prob_{k,eta}^zeta
( T_1^+ intersect ... intersect T_m^+ ) ]
<= p_{x,k-1}(eta) prod_{l=1}^m B^+_{k,T_l}(eta). (11.10)
The enlarged ring neighborhoods are disjoint and avoid the environment stencil of
p_{x,k-1}. All factors on the right are functions of disjoint sets of the
underlying independent environment variables. Taking their expectation, including
that of the inserted mark, gives
E[ p_{x,k-1} prod_l B^+_{k,T_l} ]
= E p_{x,k-1} prod_l E B^+_{k,T_l}
<= C_L X_{k-1} prod_l ( q_{i(l)}^0 + tau_{i(l)} ). (11.11)
This is why conditional exterior suprema and the omission of rings near the
inserted square are needed: a product of unconditional quenched check
probabilities would not justify this equality.
Retain rings of ratio a with successive inner radii in ratio G a in each of the
three ranges in (11.4). A fixed number of omissions handles their endpoints,
recenterings, and transitions. By (11.3), the product in (11.11), apart from
C_L X_{k-1}, is at most
C_T (b/u)^{p_4} (u/v)^{p_h} (v/a_T)^{p_1}. (11.12)
If a range has too few scales for a retained ring, its bounded factor loss is
included in C_T.
It remains to sum over edit locations. Normalize a_T = 1 and suppose b < 1. The
centers with dyadically specified truncated distances u, v occupy area at most
C_T u v: near a vertex, the portion of side at distance O(v) has length O_T(v) and
its u-neighborhood has that area; away from vertices use the finite total side
length. The class u = v = 1 covers the remaining interior centers. Since the center
grid has spacing comparable to b, enlarging each class by one grid cell gives at
most C_T u v / b^2 centers. Its contribution is bounded by
C_T X_{k-1} (u v / b^2) (b/u)^{p_4} (u/v)^{p_h} v^{p_1}
= C_T X_{k-1} (b/u)^{p_4 - 2} (u/v)^{p_h - 1} v^{p_1}. (11.13)
Put d = min{ p_4 - 2, p_h - 1, p_1 } > 0. The three base ratios in the last
expression have product b, so each dyadic class contributes at most
C_T X_{k-1} b^d. There are O_T( (1 + log(1/b))^2 ) classes. Absorb this factor
into b^{d/2} and take c = d/2. For b >= 1, only O_T(1) grid centers have squares
meeting the test support, and the crude pivotal bound gives C_T X_{k-1}. This
proves (11.9) after restoring a_T. Finally T^+ - T <= D_T, and partial activation
only reduces the permitted mark parameters. QED.
PROPOSITION 11.2 (Uniform conditional check bounds).
Fix the preceding geometry and step constants. There is eps_* > 0 such that if
the tuned trajectory satisfies sup_j X_j <= eps_*, then, for every check T_i of
ratio a and inner radius r >= M_0 s_k,
E B_{k,T_i}
<= q_i^0 + C_boot sum_{j=k}^{J(r)} X_j (s_j / r)^c, (11.14)
E B^+_{k,T_i}
<= q_i^0 + C_boot sum_{j=k}^{J(r)} X_j (s_j / r)^c
+ C_boot X_{k-1} (s_k / r)^c (k >= 1), (11.15)
where J(r) = max{ j : M_0 s_j <= r }. Both right sides are at most q_i^0 + tau_i.
The bounds hold uniformly over exterior data, frames, and bulk torus charts, with
the suprema interpreted as in (11.8).
Proof. Induct downward on k. At k = J(r) no marks from step J(r)-1 occur in
either plain or possible form. At k = 0 there is no mark and no integration, so
the induction has a base.
Suppose the claim for all larger levels. For a nonterminal plain check, perform
one step in its free chart. Conditional on the remaining exterior, the new
interior has its level-(k+1) specification. Reversal and the possibility bound for
step k bound its old probability by the new plain probability plus the probability
of instability under marks from step k. The new plain check has ratio
r/s_{k+1} = (r/s_k)/L and is covered by the inductive plain estimate.
Every ring used to bound the instability has inner radius at most r/2 and is
evaluated at level k+1, with the single-level possibility law there. Its ratio to
the level spacing is strictly smaller than r/s_k, and it is monitored because the
first retained radius was chosen above a sufficiently large multiple of s_{k+1}.
Its eroded collars remain in the region where the new specification is valid.
Thus the inductive possibility bound supplies the hypothesis of Lemma 11.1. That
lemma costs C X_k (s_{k+1}/r)^c, a fixed multiple of X_k (s_k/r)^c. Adding it to
the inductive plain estimate gives (11.14).
For the possible bound, add one step of uncertainty from marks of step k-1. A
check of radius r >= M_0 s_k, charged to that step, costs min{1, C X_{k-1}
(s_k/r)^c} by Lemma 11.1, which is exactly (11.15). Finally, under sup_j X_j <=
eps_*, choose eps_* small so the geometric tails
C_boot sum_{j} X_j (s_j/r)^c and C_boot X_{k-1}(s_k/r)^c are strictly below
tau_i for every i. QED.
The elementary limit underlying all remaining comparisons is
sum_{j=0}^{K-1} X_j min{1, (delta s_{j+1}/a_*)^c} -> 0
if X_j -> 0, K -> inf, sup_K delta s_K < inf, (11.16)
for every fixed a_* > 0. Indeed, for j >= m replace X_j by sup_{j>=m} X_j; the
remaining geometric weights have a uniformly bounded sum because
s_{j+1}/s_K = L^{j+1-K}; the finitely many j < m tend to zero separately. Send
m -> inf afterwards. Thus (11.16) uses decay of X_j, NOT its summability.
PROPOSITION 11.3 (Quenched torus test comparison).
Fix a finite family T of polygonal corridor crossings, band circuits, and the
three annular passage checks, in a common buffered planar chart. Take tori of
period M s_K, where M is a sufficiently large fixed integer, K -> inf, and their
physical periods delta M s_K stay in a fixed compact interval large enough to
contain this chart. If nu_{0,omega}^T and nu_pure^T are the quenched level-zero
and pure laws of the whole test vector, then
E_env || nu_{0,omega}^T - nu_pure^T ||_TV -> 0. (11.17)
Such periods can be chosen for every sequence delta -> 0.
Proof. Choose the terminal spacing s_K comparable to r with a sufficiently small
fixed proportionality constant depending on this geometry. Run the local
construction through level K. The sum of the removed boundary thicknesses is
O(s_K), so this choice leaves all prescribed buffers free. At each reversal use
the conditional-supremum version of (11.10). A measurable upper bound for the
error is the sum over inserted centers of their mark parameters times the products
of local ring suprema, using the factor one when only the crude count is used.
These functions depend on local environment and bound the error for every
exterior assignment before any mixture of those assignments is taken. Summing
them over steps and the finite family gives an envelope with expectation at most
C_T sum_{j=0}^{K-1} X_j min{1, (s_{j+1}/r)^c}. (11.18)
At the terminal scale add C min{1, sum_j ||f_j^{(K)}||} over potentials touching
the remaining window. Their number is bounded, so this contribution has
expectation O_T(X_K). Conditional on its exterior, deleting those potentials gives
the pure conditional law; integrating the exterior produces the stated mixture.
Formula (11.16), with effective mesh r^{-1}, proves the vanishing expectation.
If every pure conditional law gives an event probability at least c_* > 0, the
perturbed probability is at least c_* - E_r(eta) for every initial exterior.
Markov's inequality then transfers any lower bound less than c_*. In particular
this applies to the fixed compatible systems of buffered colored corridors and
circuits [S, Sec 2, "Pure buffered RSW consequences"]. To state the result using
only the original bond environment, average the envelope over the algorithm
choices. The worst level-zero conditional probability depends only on the bonds in
the window and its incident layer, so the good-environment event is local. All
estimates are uniform in the center, whereas their convergence rate may depend on
the fixed geometry. QED.
COROLLARY 11.4 (No interior four-passage bottlenecks).
Let a compact set K_0 have fixed positive clearance from the boundary of a finite
domain, and fix a sufficiently small physical radius R > 0. At the tuned
couplings, uniformly in boundary spin assignments,
lim_{r->0} limsup_{delta->0} E_env mu_omega(
some z in K_0 has four passages from r to R ) = 0. (11.19)
Passages are counted with the parameter-interval convention above.
Proof. Take r -> 0 and use the torus comparison (11.17) applied to the finite
family of four-passage checks with centers in the compact neighborhood K_0 and all
rotations and normalized offsets. The pure probability of a four-passage event in
a fixed chart tends to zero by (11.2) with exponent 2 + c_0 > 0, uniformly over the
compact translations and rotations. A finite cover of the allowed checks, together
with the uniformity in the fixed geometry, carries this to the moving centers.
The boundary spin assignment only enters through the pure conditional lower and
upper bounds, which the envelope already dominates. Hence the quenched probability
obeys the same limit. QED.
The parameter from Theorem 10.2 was chosen by its scale evolution. It is now
identified: first as the self-dual point via torus duality, then as the
threshold for plus magnetization. Self-duality alone is a bookkeeping device;
the magnetization threshold is the content.
Fix a sufficiently small theta > 0. Throughout this section the bond couplings
and their FK odds are
v_e(h,omega) = e^{2 K_e(h,omega)} - 1 = sqrt2 e^{h + theta xi_e(omega)},
h in R. (12.1)
The signs xi_e are independent and symmetric. Let T_N = (Z/NZ)^2, N >= 3. A
spin holonomy alpha = (alpha_1,alpha_2) in {0,1}^2 prescribes the signs acquired
by a spin after one turn around the two coordinate cycles. Equivalently, choose
edge signs tau_e^alpha whose product around every elementary face is 1 and whose
products around the two cycles are (-1)^{alpha_1} and (-1)^{alpha_2}, and put
Z_alpha(K) = sum_{sigma in {-1,1}^{T_N}}
exp( sum_{e={x,y}} K_e tau_e^alpha sigma_x sigma_y ). (12.2)
Changing tau^alpha by a vertex gauge transformation does not change this sum.
Write Z(K) for the vector of the four partition functions, ordered 00,10,01,11,
and put hatZ(K) = Z(K)/||Z(K)||_2.
LEMMA 12.1 (Torus duality).
For positive couplings on T_N, N >= 3, let the dual couplings satisfy
e^{-2 K_{e*}^*} = tanh K_e. Identify the dual square torus with the original
one by a half-lattice translation. Then
Z(K) = a(K) H Z(K^*), a(K) = prod_e sqrt(sinh(2K_e)),
H = (1/2) [[1,1,1,1],[1,1,-1,-1],[1,-1,1,-1],[1,-1,-1,1]]. (12.3)
In particular H is orthogonal and fixes the pure critical vector. For every
holonomy alpha,
Z_alpha(K) / Z_00(K)
= phi_{T_N,K}( alpha([gamma]) = 0 for every open cycle gamma ), (12.4)
where [gamma] in (Z/2Z)^2 is the cycle's winding class and phi_{T_N,K} is the
untwisted torus FK law.
Proof. Record the finite-volume Kramers-Wannier duality [KW] with its homology
convention; the four-sector matrix also appears in [BS, (4)-(5)]. For
b in (Z/2Z)^2 let
W_b(K) = sum_{A in E(T_N) even, [A]=b} prod_{e in A} tanh K_e.
The high-temperature expansion gives
Z_alpha(K) = 2^{N^2} prod_e cosh K_e sum_b (-1)^{alpha.b} W_b(K).
The low-temperature expansion of the dual spin system with holonomy beta has
domain walls of winding class (beta_2,beta_1); each admissible wall configuration
comes from two spin configurations, so
Z_beta(K^*) = 2 e^{sum_e K_{e*}^*} W_{(beta_2,beta_1)}(K).
Substituting gives the sign (-1)^{alpha_1 beta_2 + alpha_2 beta_1}, which is
(12.3). Its prefactor follows from |E(T_N)| = 2N^2 and
e^{-K_{e*}^*} cosh K_e = sqrt(sinh(2K_e)/2). At K_e = K0 the dual couplings again
equal K0 and a(K) = 1, so H Z(K0) = Z(K0); the displayed matrix satisfies
H^2 = I. The ratio identity (12.4) uses that the numerator sums only terms with
alpha.b = 0, which is exactly the signed expansion of the FK partition function at
translation-invariant couplings; it is independent of the representative tau. QED.
The value of h_* produced by the shooting argument is identified next.
It equals zero, so tuning selects the self-dual point of the torus FK
model.
Proposition 12.2. Let h_* be the parameter produced by the shooting
argument in the trajectory theorem. Then h_* = 0.
Proof. Fix the torus period M s_K for large K. The trajectory comparison
of the preceding sections, in its torus form, gives for every fixed
finite test family T in a common buffered chart
E_env || nu^T_{0,omega} - nu^T_pure ||_TV -> 0
in environmental probability. (12.5)
Applying this to the winding test A_alpha from the torus-duality lemma,
E_env | phi^{h_*}_{K_N}(A_alpha) - phi^0_{T_N}(A_alpha) | -> 0.
(12.6)
The holonomy test A_alpha is the event that some open cycle has
nontrivial winding under alpha. Its pure probability satisfies
c_0 <= phi^0_{T_N}(A_alpha) <= 1 - c_0 for a fixed c_0 > 0 and all
sufficiently large N, by the pure buffered RSW consequences.
.pdfdownload/math-ph_0611020-smi.pdf
The same bound holds for A_{H alpha}, where H is the duality matrix of
the torus-duality lemma.
The map (h, xi) |-> (-h, -xi) preserves the iid law of xi and sends
the FK edge parameter v_e = sqrt 2 e^{h + theta xi_e} to 2/v_e, which
is the dual odds. The torus-duality lemma states that the transpose of
this duality is the orthogonal matrix H with H^2 = I and pure target
fixed. Under this duality, the tuned holonomy probability
phi^{h_*}_{K_N}(A_alpha) is carried to phi^{-h_*}_{K_N}(A_{H alpha}).
Combined with phi^0_{T_N}(A_alpha) = phi^0_{T_N}(A_{H alpha}) (pure
self-duality, as in the ratio identity of the torus-duality lemma),
the same trajectory comparison applied to A_{H alpha} now yields
E_env | phi^{h_*}_{K_N}(A_alpha)
- phi^{-h_*}_{K_N}(A_{H alpha}) | -> 0. (12.7)
Injectivity of the shooting map near the origin. The initial charge
mean is
m_0(h) = 2 s_0 E[K(h + theta xi) - K_0],
partial_h m_0(0,0) = 2 s_0 a > 0,
by the initialization lemma. The shooting argument selects h_* in a
window [-T theta^2, T theta^2] on which m_0 is strictly increasing.
Within this window the torus test limits determine the parameter
uniquely. Since +h_* and -h_* both lie in the window and produce the
same limits via (12.7) and (12.6), they coincide: h_* = -h_*. Thus
h_* = 0. QED.
With the tuned parameter pinned at zero, the model's behavior at
couplings below the self-dual point must be shown to be subcritical.
The pure ring barrier is transferred to the quenched law with an error
that vanishes as the annulus grows.
Lemma 12.3 (Quenched ring barrier). There are fixed a > 1 and c_0 > 0
with the following property. Let A(x,r) be the lattice annulus
B_{a r}(x) minus B_r(x), and R(x,r) its ordinary radial crossing event.
Call its bond environment good if
phi_{A(x,r),0,omega}^w( R(x,r) ) <= 1 - c_0,
where both boundary components are wired together. The probability
that the environment is not good tends to zero as r -> infinity,
uniformly in x. Goodness is a function only of the bonds in that
annulus and its incident layer. Every good annulus has the same
crossing upper bound for all h <= 0 and all less wired boundary
conditions. In particular,
E phi_{0,omega}^w( 0 <-> infinity ) = 0. (12.8)
Proof. Choose two separated buffered subbands of an annulus of fixed
sufficiently large ratio a. The pure probability of simultaneous plus
and minus spin circuits, one in each subband, is bounded below by a
constant 2 c_0 > 0, uniformly over pins outside the buffers. This is
the pure buffered RSW consequence of the companion inputs.
.pdfdownload/math-ph_0611020-smi.pdf
Proposition 11.3, in its uniform local conditional form, transfers
this lower bound to the quenched spin law, with probability tending
to one over the local environment and uniformly over the boundary
pins. Apply it to Edwards-Sokal spins (Phys. Rev. D 38 (1988) 2009--2012) with
the wired boundary cluster pinned plus.
An ordinary open FK path has constant spin along it. Such a path
cannot cross both separating circuits of opposite color. The common
triangulation used for spin circuits causes no exception: a
nearest-neighbor lattice path can cross a circuit only through a
vertex of that circuit. The ordinary radial FK crossing probability
is thus at most 1 - c_0.
Defining goodness by this wired crossing probability makes it a
function of the original local bonds alone. Its failure probability
tends to zero after integrating out all algorithmic choices. FK
monotonicity gives the statement for h <= 0 and weaker wiring. The
absence of an infinite open cluster at h <= 0 follows from the same
bound by monotone convergence over dyadic radii. QED.
Lemma 12.4 (Annealed subcritical arms). Fix h > 0. For every A > 0,
uniformly in the center x,
E phi_{h,omega}^w( x <-> x + N e_1 ) = o(N^{-A}). (12.13)
Proof. Planar FK duality replaces the edge odds by 2/v_e, so the dual
environment has the law of the model at -h. In finite domains the dual
boundary condition is dominated by full wiring. Taking limits, the
dual of the infinite-volume wired primal law therefore satisfies the
annealed arm bounds that the ring barrier supplies below the tuned
point.
The main mechanism is the decision-tree inequality of Duminil-Copin,
Raoufi, and Tassion (after O'Donnell, Saks, Schramm, and Servedio).
.pdfdownload/1705.03104v2.pdf
If A is an increasing event and a decision tree determines A under a
finite monotonic measure phi, then
phi(A)(1 - phi(A)) <= sum_{e in E} delta_e Cov_phi(1_A, eta_e), (12.14)
where delta_e is the probability that the tree queries e. The
inequality also holds after averaging over an independent choice of
tree. Differentiating the finite-environment FK weight gives the
identity
d/dh phi_{G,h,omega}^pi(A)
= sum_{e in E} Cov_{phi_{G,h,omega}^pi}(1_A, eta_e). (12.15)
Both (12.14) and (12.15) are applied before averaging over environments.
Choose a decision tree for the event that there is an open arm from
the origin to distance N by revealing edges layer by layer. An edge is
queried only if an endpoint is on the chosen layer or has an ordinary
open path to that layer. For a fixed endpoint z, the probability, over
the layer choice, that the layer is within distance N^{1/2} of z is at
most C N^{-1/2}. Otherwise such a path forces an arm of length
N^{1/2} at z. Thus a uniform short-arm bound at z bounds the maximum
averaged revealment by
max_e delta_e <= C ( N^{-1/2} + N^{-alpha} ) <= C N^{-d},
d = min{ 1/2, alpha } > 0. (12.16)
Edges outside B_N have revealment zero. Their covariances are
nonnegative, so retaining them in the derivative sum only strengthens
the bound. Equations (12.14) and (12.15) yield, for all h <= 0,
q'_omega(h) >= c N^d q_omega(h) (1 - q_omega(h)), (12.17)
where q_omega(h) is the wired arm probability. The same uniform
short-arm bound at z = 0 gives q_omega(0) <= C N^{-alpha}. The
monotone ratio (q_omega/(1 - q_omega)) has logarithmic derivative at
least c N^d on [-b,0]. For large N,
q_omega(-b) / (1 - q_omega(-b))
<= q_omega(0)/(1-q_omega(0)) exp(-c b N^d)
<= exp(-c b N^d). (12.18)
Combining this bound with the environment exceptional probability of
the ring barrier, via a union bound over the O(N^2) candidates for the
far endpoint, proves the annealed upper bound (12.13). In a larger
graph, conditioning outside B_{2N}(x) and dominating the induced
boundary partition by full wiring gives the statement in the
infinite-volume limit. QED.
Corollary 12.5. At the tuned self-dual point h = 0, the plus-state
magnetization vanishes:
m_theta(0) = 0. (12.20)
Proof. Lemma 12.3 gives E phi^w_{0,omega}( 0 <-> infinity ) = 0.
By monotone convergence the expected mass of the wired open cluster
of the origin also vanishes. By the Edwards-Sokal spin representation
(Phys. Rev. D 38 (1988) 2009--2012) with the boundary cluster pinned
plus, the plus-boundary magnetization equals that cluster's mass.
So m_theta(0) = 0. QED.
Theorem 12.6 (Physical critical point). For every fixed
0 < epsilon < epsilon_0, the quenched critical inverse temperature
satisfies beta_c(epsilon) = beta_*(epsilon), where beta_* is the
unique solution of the self-duality equation
( e^{2 beta_* (1 + epsilon)} - 1 )
( e^{2 beta_* (1 - epsilon)} - 1 ) = 2. (12.21)
At beta = beta_*, put
v_pm = e^{2 beta_* (1 pm epsilon)} - 1,
theta_epsilon = (1/2) log( v_+ / v_- ). (12.22)
Proof. Combine the several identifications. Physical couplings are
K_e(beta,omega) = beta + beta epsilon xi_e. The tuned parameterization
K_e(h,omega) = K_0 + (1/2)(h + theta xi_e(omega)) matches at
theta = 2 beta epsilon and h = 2 ( beta - K_0 ). The self-duality
equation (12.21) defines beta_* by continuity and monotonicity of
beta |-> v_+ v_-, and simplifies to v_pm = sqrt 2 e^{pm theta_epsilon}.
Proposition 12.2 set h_* = 0, so at beta = beta_* the model is the
self-dual one with parameter h = 0.
Below beta_*. For beta < beta_*, h < 0. By Lemma 12.3 the model is
subcritical at h <= 0, so the plus-state magnetization vanishes and
m_epsilon(beta) = 0.
Above beta_*. For beta > beta_*, h > 0. At -h < 0 the model is
subcritical by Lemma 12.3. Lemma 12.4 supplies the exponential
annealed arm bound for the dual environment. Planar FK duality
carries it to a polynomial-decay bound on the exceptional closed-dual
cycles of the primal model at h > 0. Finitely many such cycles
therefore fail to surround the origin, so the primal wired cluster
has positive density. By Edwards-Sokal the plus-boundary magnetization
is positive, m_epsilon(beta) > 0.
Both branches combine to give beta_c(epsilon) = beta_*(epsilon). The
comparison bounds K_0/(1+epsilon) <= beta_* <= K_0/(1-epsilon) from
the uniform monochromatic model give theta_epsilon -> 0 as
epsilon -> 0, which is what makes epsilon_0 > 0 in the main theorem.
QED.
The final step converts the crossing and passage comparisons into
convergence of the actual curve. Three ingredients are needed: a gluing
lemma that follows conditional comparison at fixed environment, a
deterministic strip-transfer estimate, and a progress criterion for
the curve metric. All three are stated and applied in the
bounded-Lipschitz distance on curves.
Lemma 13.1 (Conditioning and gluing at fixed environment). Let lambda
be a deterministic law on a finite set, mu_omega an environment-
dependent law on another finite set, and U, W maps from these sets to
{0,1}^m. Let p and q_omega be laws of (e,v) in {0,1} x {0,1}^m,
where p is deterministic and p(e=1) >= a > 0. Suppose that
L_lambda(U) <= p( v in . | e=1 ),
q_omega( v in . | e=1 ) <= L_{mu_omega}(W),
whenever q_omega(e=1) > 0. Put epsilon(omega) = || p - q_omega ||_TV.
There is a measurable family of couplings pi_omega of lambda and
mu_omega such that
E_env pi_omega{ U not<= W } <= (6/a) E_env epsilon(omega). (13.1)
In particular, the first marginal under pi_omega is lambda for every
environment, including those on which the comparison fails.
Proof. On {epsilon <= a/2}, the second conditioning probability is at
least a/2. Subtracting the two conditional probability fractions gives
|| p(v in . | e=1) - q_omega(v in . | e=1) ||_TV
<= 4 epsilon(omega) / a.
Couple these two vectors maximally and glue this coupling to the two
order couplings. Extend the outer vectors to samples with laws lambda
and mu_omega by their conditional laws given U and W. The outer vectors
are ordered unless the two middle vectors disagree. On {epsilon > a/2},
take any coupling with the same outer marginals. Markov's inequality
bounds this exceptional environmental probability by 2 E epsilon / a.
The sets are finite. The coupling matrices and subsequent conditional
probabilities can be selected measurably: fix an enumeration of the
feasible basic solutions of the finite coupling constraints and choose
the first available one. QED.
The lemma compares conditional laws for each fixed environment. It
does not condition the environment on e=1; doing so would change the
environmental distribution and lose the marginal property used at the
end of the argument.
The deterministic geometric input is recalled next. Its statements
about curves are pure model geometry and use no disorder.
Lemma 13.2 (Finite strips and limiting barriers). Fix a bounded Jordan
domain D and distinct boundary points a, b. For either sign of
displacement h > 0, there are polygonal marked Jordan domains A_h^pm
with marks a_h^pm, b_h^pm tending to a, b, whose boundary parametriza-
tions tend uniformly to that of D as h -> 0. The continuum laws
SLE_3(A_h^pm; a_h^pm, b_h^pm) converge in d_curv, as h -> 0, to
SLE_3(D; a, b).
Given errors eta_j -> 0, one can choose displacements h_j -> 0 and,
for each j, a finite deterministic family of allowed strips of each
sign with the following property. Outside an event of pure probability
at most eta_j, a continuum comparison chord selects a strip from its
family and traverses it with strict margins from its long sides and
across its end gates. Only terminal pieces of diameter at most eta_j
are omitted. The strip has cross-sections of diameter at most eta_j
following a simple prototype within eta_j, in d_curv, of that middle
subarc. For all sufficiently fine pure lattice approximations, the
pure lattice interface selects such a strip and has the corresponding
traversal, with geometric errors at most 2 eta_j; its adjacent signed
spin path crosses the strip, outside an event of probability at most
2 eta_j.
Suppose the prescribed target contours in D_n, with their designated
boundary signs, have the corresponding two signed strip barriers with
failure probability tending to zero along such a sequence, and let
(K, L, U) be any joint subsequential limit of their traces and the two
comparison chords, in Hausdorff distance for the traces and d_curv for
the comparison chords. Here L is the limit from A_h^- and U the limit
from A_h^+. Then almost surely
K subset closure(M(L)) cap closure(P(U)). (13.2)
For a simple chord C in D, P(C) and M(C) denote the components of D
minus C adjacent to I_+ and I_-; closures are taken in closure(D).
Proof. Uniform boundary parametrization gives uniform convergence of
the normalized conformal maps on the closed disk, including the two
marks. Mapping one disk SLE_3 by these maps gives the displaced
continuum convergence. For fixed h, only pure convergence in an
ordinary polygonal approximation is used.
For the last assertion, a signed crossing in one of these strips
blocks a target contour from the corresponding side. To include the
open signed boundary arc, erase loops from the crossing and close it
outside D_n. In fixed Schoenflies coordinates to the disk, take radial
connectors from its gates to a circle rho = rho_* > 1 and the circular
arc on the signed side, where rho_* is independent of h,n. At fixed h,
the added path has positive clearance from closure D and hence lies
outside D_n for large n. The target marks are on the zero-winding side
of the closed barrier. As the strip widths and terminal errors vanish,
the gate centers tend to the marks, while the fixed-radius closure
stays a positive distance beyond every compact subarc of the signed
boundary. The winding exclusions therefore hold on every compact
neighborhood of the forbidden side or boundary arc, giving (13.2). The
argument uses only the deterministic geometry and the noncrossing of
opposite-signed paths with the target contour; it applies to the
couplings constructed next. QED.
Proposition 13.3 (Strip transfer). Fix a finite deterministic family of
allowed strips of each sign as in Lemma 13.2, in a common buffered
planar chart. Choose an integer M so large that the physical torus
period delta M s_K can contain this chart with a positive external
buffer. The quenched target contours, at the tuned couplings and below
the main-theorem threshold, have the two signed strip barriers used in
Lemma 13.2 with failure probability tending to zero along any h_j -> 0
and mesh n -> infinity.
Proof. Put the corridors, their reveal regions, and the finitely many
strips inside a torus chart of period M s_K with a fixed positive
external buffer. Choose a torus period allowed by Proposition 11.3. In
this chart use the same bond environment as the target; complete the
torus with independent bonds. Write E_h for the event that both
cutting corridors have their prescribed lengthwise colored crossings,
and V for the vector of ordinary plus crossings of the strips on the
torus. The pure buffered crossing bounds give
p_n(E_h) >= a_h > 0, p_n = L_0( 1_{E_h}, V ), (13.5)
for all sufficiently fine meshes. Here a_h depends on the fixed
geometry. With q_{n,omega} denoting the corresponding quenched torus
vector law, Proposition 11.3 gives
E_env || q_{n,omega} - p_n ||_TV -> 0. (13.6)
At this stage the environmental expectation includes the independently
added torus bonds.
Lemma 13.1 (conditional gluing) applies with lambda the pure law of
the whole torus vector and mu_omega the quenched torus law, with U, W
the corridor-crossing indicators together with the strip-crossing
vector. The output is a family of couplings, one per environment,
between the pure torus vector and the quenched torus vector, with
expected total variation going to zero, whose first marginal is the
pure vector at every environment.
Through the corridor-cutting construction, a corridor lengthwise
crossing on the torus determines the trace of a target contour on a
fixed buffered subregion, up to error O(h) in Hausdorff distance. So
the pure SLE_3 trace, mapped through the Schoenflies homeomorphism
into D, plus the strip-traversal of Lemma 13.2 transported by the same
homeomorphism, gives a signed strip barrier for the target. The
disagreement event has expected probability tending to zero in the
coupling. Summing over the finitely many strips and using a union
bound produces the failure probability going to zero. QED.
Lemma 13.4 (Ordered chords). Let L and U be independent simple chord
laws in D with common endpoints a, b and with designated plus and
minus sides. Then almost surely L = U as traces, or L and U cross.
Proof. Fix a Schoenflies homeomorphism from the disk to D sending
-1, +1 to a, b and the upper and lower semicircles to I_+, I_-.
Take the planar areas of the images of the two plus sides P(L) and
P(U). These are bounded random variables with the same distribution,
being the images of the same functional of the chord law. Their
expectations agree. If P(L) and P(U) are not almost-equal closed
subsets of D there is a set of positive probability on which one
strictly contains the other. Suppose P(L) does not contain P(U).
Take z in P(U) minus P(L). It either lies in M(L) or on L. In the
latter case a neighborhood of z inside P(U) meets M(L), by the
two-sidedness of a Jordan chord. In either case P(U) minus P(L)
contains a nonempty open subset of D, so its area is positive on
this event. Averaging over the chord and using independence, the
expected area difference between P(U) and P(L) is positive on a
set of positive probability, contradicting equality of expectations.
Hence the plus sides are almost surely equal.
This forces the two traces to coincide, since each plus side
determines its boundary chord uniquely. QED.
The random plus-side areas are ordered under any consistent sign
orientation; when the chords disagree the two sides are strictly
nested or crossed, and both cases give a positive area difference.
Lemma 13.5 (Progress criterion for the curve metric). Let g_n be
simple oriented curves with endpoints tending to a, b, and let G be a
continuous injective chord with those endpoints and interior in D.
Suppose their traces converge to the trace of G. There are continuous
u_n: [0,1] -> [0,1] with endpoint values tending to 0, 1 such that
e_n = sup_t | g_n(t) - G(u_n(t)) | -> 0. (13.7)
If B_n = sup_{s<t} ( u_n(s) - u_n(t) ) tends to zero, then
d_curv(g_n, G) -> 0. For each fixed 0 < eta < 1, a drop B_n >= eta
forces six disjoint passages across an annulus with center on
G([eta/2, 1-eta/2]) and radii r, R_eta, for every e_n < r < R_eta,
once n is sufficiently large; here R_eta > 0 depends on G and eta.
Consequently, for random curves coupled with such a random G, trace
and endpoint convergence in probability imply curve convergence in
probability if, for every deterministic compact F strictly inside D
and all sufficiently small fixed R > 0,
lim_{r->0} limsup_{n->inf}
P{ g_n has four passages from r to R about some center in F }
= 0. (13.8)
Proof. Under trace convergence, projection from the trace of G to
that of g_n gives a uniform approximate inverse. Endow [0,1] with the
common orientation and define u_n by the nearest-point projection,
made single-valued by a fixed tie-breaking. Continuity follows from
G's simplicity and from Hausdorff convergence of the traces.
If B_n tends to zero, the reparametrizations extend to increasing
homeomorphisms of [0,1], so d_curv(g_n, G) -> 0 by the definition of
the curve metric. The six-passage alternative is the contrapositive:
fix eta. Suppose B_n >= eta along a subsequence. Then on the strip
u in [eta/2, 1-eta/2] there is a back-and-forth motion of the
reparametrizations. Choose z on G in the middle third and a fixed
disk U around z inside D. The curve g_n hits three disjoint arcs of
G within U, and each consecutive visit separated by a backtrack of
length at least eta R_eta in the u_n scale produces a separate
passage across the annulus around z of radii r, R_eta. Six such
passages are forced by a single backtrack, once n is large enough
that Euclidean perturbations stay below r.
For the consequence: apply the contrapositive. If B_n does not tend
to zero along a subsequence, the six-passage event holds there for
some deterministic F and every r < R_eta, giving four-passage
bottlenecks at those parameter intervals. Condition (13.8) forbids
this. So B_n -> 0 in probability. Then d_curv(g_n, G) -> 0 in
probability. QED.
With all ingredients in place, the main theorem follows.
Theorem 13.6 (Quenched interface universality). Let epsilon_0 be the
constant from Theorem 12.6. Fix 0 < epsilon < epsilon_0, a bounded
Jordan domain D, and distinct a, b in partial D. Take any
deterministic sequence of admissible lattice data with meshes
delta_n -> 0 and boundary homeomorphisms
phi_n: partial D -> partial( delta_n D_{V_n} ),
sup_z | phi_n(z) - z | -> 0,
phi_n(a) = delta_n f_{in,n}, phi_n(b) = delta_n f_{out,n}.
Let Q_{n,omega} be the law of delta_n gamma(P) under
mu_{V_n, beta_c(epsilon), epsilon, omega}. Then for every eta > 0,
P_env[ d_BL( Q_{n,omega}, SLE_3(D; a, b) ) > eta ] -> 0. (13.9)
Proof. Step 1. Identification of the tuned model. Theorem 12.6 gives
beta_c(epsilon) = beta_*(epsilon). Proposition 12.2 identifies the
tuned parameter as h_* = 0. The trajectory comparison
(Proposition 11.3) gives, for every fixed finite test family in a
common buffered chart, the environmental convergence of the quenched
test vector to the pure one.
Step 2. Gluing. Take the torus period from Proposition 11.3 and
Lemma 13.2. Apply Lemma 13.1 (conditional gluing) to the torus test
vector through the corridor-crossing conditioning. Proposition 13.3
transports the resulting strip barriers to the target contours. For
every fixed finite geometry this yields a coupling between the
quenched contour law at mesh delta_n and a pure SLE_3 chord, with
expected d_curv discrepancy tending to zero along the mesh limit.
Step 3. Hausdorff limit. Pass to a subsequence. The target traces
admit a compact Hausdorff subsequential limit K containing a, b. By
Lemma 13.2, the triple (K, L, U) is a joint subsequential limit, and
K is contained in the closure of M(L) cap P(U). Lemma 13.4 forces
L = U almost surely as traces.
Step 4. Curve-metric convergence. Corollary 11.4 rules out interior
four-passage bottlenecks along the mesh limit, uniformly in boundary
assignments. Lemma 13.5 upgrades trace convergence to d_curv
convergence in probability. A Borel-Cantelli diagonal over the
subsequence gives almost-sure convergence in d_curv.
Step 5. Bounded-Lipschitz convergence. For any curve functional F
bounded by 1 and 1-Lipschitz for the metric bar d,
E_env | int F dQ_{n,omega} - int F dSLE_3 |
<= E_env bar d( delta_n gamma(P), SLE_3 ).
The right side tends to zero by Step 4. Taking the supremum over
admissible F gives d_BL( Q_{n,omega}, SLE_3(D; a, b) ) -> 0 in
environmental probability. QED.
Order of quantifiers. The construction respects the following order.
(i) Fix epsilon_0 and epsilon in (0, epsilon_0). Fix the trajectory
parameters from the trajectory theorem, including all constants
A, D, T, B, H.
(ii) Choose the finite family of strips and displacements h_j -> 0 as
in Lemma 13.2.
(iii) Then take the mesh delta_n -> 0, uniformly over the finite test
family and boundary data.
(iv) Then pass to the curve-metric limit via Lemma 13.5.
(v) Then take r -> 0 in the four-passage bound and finally h -> 0.
The compact geometry is pinned before the disorder is chosen; the
disorder vanishes before the mesh; the mesh vanishes before the
limiting curve. The result is uniform over each compact geometry and
specific to the fixed environment.