diff 56bc728..f6bf760 o4_model_mass_bounds_v2.txt
diff --git a/o4_model_mass_bounds_v2.txt b/o4_model_mass_bounds_v2.txt
index 4de2c60..e84dfed 100644
--- a/o4_model_mass_bounds_v2.txt
+++ b/o4_model_mass_bounds_v2.txt
@@ -66,187 +66,10 @@ positive full gap at every finite beta>0. Whether the lattice theories
converge to a continuum quantum field theory, and what its exact spectrum
there looks like, is not decided here; we state separately the
extra convergence hypotheses that would turn our lattice bounds into a
-mass-gap statement for a continuum theory. The proof is the one of the
-original technical paper; this version reorganises it and adds plain-English
-commentary, taking the stated core estimates on faith where necessary.
+mass-gap statement for a continuum theory. The proof is that of the original technical paper.
-vspace*{-12pt}
-tableofcontents
-newpage
-%---------------------------------------------------------------------
-# Overview: what is being proved and how
-
-%---------------------------------------------------------------------
-
-## The model in one paragraph
-
-Place a spin sigma_xin S^3subsetR^4 at every site x of a large
-periodic square lattice Λ=(Z/nZ)times(Z/wZ). Give the spins
-the Boltzmann weight
-
- ddmu_{beta;n,w}(sigma) = Z_beta(n,w)^{-1}
- exp!(beta sum_{< xy>}sigma_xcdotsigma_y)
- prod_{xinΛ}ddomega(sigma_x),
-
-where < xy> runs over nearest-neighbor pairs, each counted
-once, ddomega is the normalised uniform measure on S^3, and
-Z_beta(n,w) is the normalisation. This is the two-dimensional O(4)
-nonlinear sigma model on the lattice with nearest-neighbor couplings and
-bare inverse coupling beta.
-
-The energy betasigma_xcdotsigma_y = tfracbeta2 - tfracbeta2
-|sigma_x-sigma_y|^2 favours neighbouring spins pointing in the same
-direction, and beta is how strongly. There is no spin-ordering
-transition in two dimensions (Mermin--Wagner), but there is a length
-scale beyond which correlations die exponentially. That length scale is
-what we want to bracket.
-
-## What ``mass gap'' means
-
-On a cylinder of spatial width w the transfer operator K_w advances
-the configuration one time step. Its kernel is
-
- K_w(s,s')=exp!( (beta)/(2)V_w(s)
- +betasum_{i=1}^w s_icdot s'_i
- +(beta)/(2)V_w(s') ),
- V_w(s)=sum_i s_icdot s_{i+1}.
-
-K_w is a positive, self-adjoint, compact integral operator. Its largest
-eigenvalue lambda_1 is simple and its positive ground eigenvector is
-the vacuum. Closing time periodically gives Z_beta(n,w)=Tr K_w^n.
-
-Fix beta, take wto∞, and take a subsequential local limit. On
-the resulting reflected Hilbert space there is a positive one-step
-translation contraction T_beta with T_betaΩ = Ω. The
-full transfer gap is
-
- gap(beta) = -lognorm{T_betabig|_{Ω^perp}}.
-
-Equivalently, autocorrelations < F, T_beta^d F> of every
-centred local observable F decay at least as fast as e^{-gap d},
-where d is the time separation. The gap is the mass; gap=0 means
-critical or ordered.
-
-The subtlety: controlling the spin two-point function is not enough.
-There is a second, rotation-invariant sector of observables -- the
-bond energies sigma_xcdotsigma_y -- and T_beta has spectral
-support in that sector as well. The full gap includes both. Our
-theorem bounds this bigger, smaller gap. That is why the analysis has to
-carry along every local observable, not just the spin.
-
-## The answer
-
-[Main result]
-There are finite constants beta_0,c,C, with 0<c<C, such that for
-every betagebeta_0 the periodic measures (eq:model-ov) have a
-unique local limit, and its full transfer gap obeys
-
- c,sqrtbeta,e^{-pibeta};le;gap(beta);le;
- C,sqrtbeta,e^{-pibeta}.
-
-
-
-Three comments on the formula sqrtbeta,e^{-pibeta}.
- The exponent is not a guess; it is forced by the renormalisation
- group (RG). Asymptotic freedom says the running coupling satisfies
- betamapstobeta - gamma - gamma/(2pibeta)+... with
- gamma=(log L)/pi, and integrating over many scale factors gives
- log(inverse length)simpibeta. So it is a two-loop
- calculation that produces e^{-pibeta} with a sqrtbeta
- correction.
- Up to bounded positive factors, sqrtbeta,e^{-pibeta} is
- the natural asymptotically free scale for this model in these
- units. That is why the bounds are sharp.
- The bounds are not asymptotic (the ratio need not tend to 1, and
- in fact the exact O(4) prefactor is a known transcendent constant
- we do not touch here). They give the correct order of magnitude.
-
-
-[Full gap at every finite beta]
-For every finite beta>0, the periodic measures on squares have a unique
-local limit. Its full transfer gap is positive, with
-[
- gap(beta);ge; c,exp!(-pibeta
- - C√((1+beta)log(2+beta)))>0
-]
-for some absolute constants c,C>0.
-
-
-Corollary~cor:allT-ov does not use the sharp two-loop
-comparison; it follows from a preliminary mixing estimate alone. The
-gap is therefore positive at every temperature, but the formula is too
-weak in the exponent to distinguish sqrtbeta,e^{-pibeta} from
-beta^{-1/2}e^{pibeta+sqrtbeta}. Theorem~thm:main-ov repairs
-that at large beta.
-
-## From gap to a physical mass
-
-Choose a physical reference length ell_{ref}>0 and declare
-that L^N lattice spacing (where L is a fixed block scale we will
-choose later) has physical length ell_{ref}. The lattice
-spacing is a_N=ell_{ref}L^{-N}. Then
-gap(beta_N)/a_N is the mass of the cutoff theory in physical units.
-Theorem~thm:main-ov upgrades to
-
- frac{log 2}{M_0ell_{ref}}
- ;le; (gap(beta_N))/(a_N)
- ;le; frac{log 8}{ell_{ref}}
- (N large),
-
-with M_0 a fixed positive integer and beta_N chosen so that exactly
-N RG steps have been done at terminal coupling H. Bounded positive
-multiplicative factors sandwich the continuum-normalised mass. Whether
-the cutoff measures converge to an actual continuum theory is a separate
-question; we spell out in Section~sec:physical the additional
-hypotheses that would let one read (eq:physical-main-ov) as a
-statement about an existing continuum spectrum.
-
-## Relation to earlier work, in one paragraph
-
-Absence of ordinary ordering is classical (Mermin--Wagner, and the
-McBryan--Spencer quantitative bounds). Exponential decay of the spin
-correlation was proved separately for every finite beta and every
-internal dimension ge 3. Perturbative renormalisation predicts the
-scale sqrtbeta,e^{-pibeta}; Bethe ansatz S-matrix theory fixes
-the exact continuum mass for O(3) and O(4), and for O(n) in
-general. None of that gave a full transfer gap (including invariant
-observables) for the specified lattice model. Our theorem closes that
-gap for O(4) at large beta, and yields positivity at every finite
-beta as a corollary. The proof of the exact continuum mass
-(including the ``Gamma-function'' prefactor) is a separate problem,
-and the proof of lattice-to-continuum convergence is also separate.
-
-## The proof in six steps
-
-The proof has a clear overall shape.
-{bf Step 1} (Section~sec:finite-volume). Reduce the gap to a
-finite-volume partition-function defect
-Delta_beta(n)=4log Z_beta(n,n)-log Z_beta(2n,2n).
-{bf Step 2} (Section~sec:preliminary). Show Delta_beta(n) is
-already small at some scale nsim e^{pibeta+C√(betalogbeta)},
-using Ward identities and a stiffness iteration.
-{bf Step 3} (Sections~sec:free-gap-ov-run--sec:calibration).
-Construct an exact real-space renormalisation group (RG) step that
-integrates out Ltimes L blocks and rewrites the coarse lattice in the
-same controlled form. Show it contracts past a coupling of size Hsim
-constant.
-{bf Step 4} (Section~sec:trajectories). Show that, at every
-depth N, one can pick a bare coupling beta_N so the N-step
-trajectory ends exactly at H/c. Matching at the coarse end absorbs
-the difference between two different depths.
-{bf Step 5} (Section~sec:comparison). Compare the positive
-coarse densities (partition functions/correlation functions) rather
-than the signed expansions; positivity of the original model does the
-hard work.
-{bf Step 6} (Sections~sec:assembly--sec:upper). Choose one
-reference box once at a coarse depth K, show Step 2 already makes
-Delta small there, and Step 4 keeps it small at every finer cutoff.
-Then apply Step 1 to get the lower mass bound. Prove the upper bound by
-noticing that the RG produces a definite correlation at distance L^N.
-
-The rest of the paper expands each step.
%---------------------------------------------------------------------
# The model and the full transfer gap
@@ -263,13 +86,13 @@ nearest-neighbour bond is counted once.
Fix a spatial circumference w. A time-slice configuration is
s=(s_1,...,s_w)in(S^3)^w; the one-step interaction is
-V_w(s)=sum_{i=1}^w s_icdot s_{i+1} with periodic indices. The transfer
+V_w(s)=sum_{i=1}^w s_i· s_{i+1} with periodic indices. The transfer
operator K_w has kernel (eq:transfer-kernel-ov). It is:
compact and self-adjoint: beta finite, S^3 compact,
kernel continuous;
positive semidefinite: expand
- e^{beta scdot s'}=sum_m beta^m (scdot s')^m/m! and each
- (scdot s')^m is a positive tensor-product kernel, then sandwich
+ e^{beta s· s'}=sum_m beta^m (s· s')^m/m! and each
+ (s· s')^m is a positive tensor-product kernel, then sandwich
with e^{beta V_w/2};
trace class: the diagonal integral of the positive kernel
series converges;
@@ -326,26 +149,13 @@ Section~sec:calibration makes that identification precise.
The rest of the paper is the proof.
-## Six-step map, with cross-references
-
-
-Step 1: Delta_beta criterion dotfill Ssec:finite-volume
-Step 2: preliminary mixing dotfill Ssec:preliminary
-Step 3: linear block dotfill Ssec:linear
-Step 3': exact nonlinear RG dotfill Ssec:rg
-Step 3'': calibration dotfill Ssec:calibration
-Step 4: admission and matching dotfill Ssec:trajectories
-Step 5: positive-density comparison dotfill Ssec:comparison
-Step 6: assembly and upper bound dotfill Ssec:assembly, Ssec:upper
-Physical units dotfill Ssec:physical
-
## Notation
gap(beta) = full transfer gap. rho(beta)=sqrtbeta e^{-pibeta}
= asymptotically free inverse length. L = blocking factor (dyadic,
large). gamma=(log L)/pi. Λ_{n,w}=(Z/nZ)times(Z/wZ);
v=|Λ_{n,w}|. Z_beta(n,w) = partition function. E = expectation.
-norm{cdot} = operator or supremum norm, context clear. C,c>0 =
+norm{·} = operator or supremum norm, context clear. C,c>0 =
generic constants that may change between formulas but are
beta-independent unless stated.
@@ -395,7 +205,7 @@ q(n,w)ge0.
Introduce the normalised trace weights
t_i = (lambda_i(w)^n)/(sum_jlambda_j(w)^n)
- qquadsoqquad
+ so
e^{-p(n,w)} = sum_i t_i^2.
The right side is the probability that two independent copies of the
@@ -581,7 +391,7 @@ Throughout, spins take values in the unit sphere S^3subsetR^4,
identified with the unit quaternions, with normalised Haar measure.
Distances between supports use maximum norm (or its periodic version).
-## What we will prove
+## Preliminary mixing theorem
[Preliminary mixing]
There are absolute constants C,c,p>0 such that, with
@@ -667,10 +477,10 @@ The rest of the section proves each stage.
Work on a pinned square Q(S)={0,...,S}^2. For each positively
oriented edge e=xto y, write
- t_e + s_e = q_y,U_e,q_x^{-1}, t_einR, s_einImmathbb H,bbH,
+ t_e + s_e = q_y,U_e,q_x^{-1}, t_einR, s_einImH,
where U_e is a fixed unit quaternion encoding a possibly twisted edge.
-The scalar t_e=q_ycdot q_x is the bond energy up to twist, and the
+The scalar t_e=q_y· q_x is the bond energy up to twist, and the
imaginary quaternion s_e is the ``current'' along the edge: its three
components measure the three perpendicular rotation axes. The pinning is
encoded by taking a fixed boundary spin.
@@ -683,13 +493,13 @@ Z^{-1}exp(sum_e b_e t_e).
For a 3times2 matrix-valued macroscopic test field F_e on edges,
define the (macroscopic) current
- X_Q(F)=S^{-1}sum_e b_e,s_ecdot F_{mu(e)}(z_e),
+ X_Q(F)=S^{-1}sum_e b_e,s_e· F_{mu(e)}(z_e),
where z_e is the edge midpoint and mu(e) the edge direction. Its
second variation under infinitesimal spin twists gives the stiffness
session
- mathcal H_Q(F,G)=S^calH_Q(F,G)=S^{-2}Esum_e b_e t_e,barbar F_eF_ecdot G_e
+ H_Q(F,G)=S^calH_Q(F,G)=S^{-2}Esum_e b_e t_e,barbar F_eF_e· G_e
-Cov(X_Q(F),X_Q(G)).
This is ``direct quadratic cost minus current covariance''. The
@@ -709,8 +519,8 @@ unit imaginary a changes not only the (1,a)-plane but also the
perpendicular imaginary plane, with a cross-product correction. The
concrete formulas are:
- D_fmathcal AcalA &= sum_e b_e(s_ecdot a)(f_y-f_x), notag
- D_f^2mathcal AcalA &= sum_e b_e t_e(f_y-f_x)^2
+ D_fA &= sum_e b_e(s_e· a)(f_y-f_x), notag
+ D_f^2A &= sum_e b_e t_e(f_y-f_x)^2
for a one-quaternion-axis profile a f_x, and the exact nonabelian
@@ -726,7 +536,7 @@ The Ward identity expressing invariance of the measure under rotations
by eta (with eta=0 on the pinned boundary) has the form
E[barbar X_Q(d_SX_Q(d_Seta),X_Q(F)]
- = S^{-2}Esum_e b_e t_e,(d_Seta)_ebarcdot F_e
+ = S^{-2}Esum_e b_e t_e,(d_Seta)_ebar· F_e
+ E X_Q!(F_etimes(bareta_y+bareta_x)).
The left side is a current-correlation; the first term on the right is
@@ -750,11 +560,11 @@ Suppose throughout log(2+S)le16B.
Proof.
Consider first a compact score f (a one-axis profile supported in a
-patch, zero on the boundary). Taylor's formula for D_f^2mathcal AcalA is
+patch, zero on the boundary). Taylor's formula for D_f^2A is
bounded in absolute value by bsum_e(f_y-f_x)^2, so Haar invariance
gives
- Eexp(tD_fmathcal A)calA)
+ Eexp(tD_fA)
le exp!Bigl[tfrac12 bt^2sum_e(f_y-f_x)^2Bigr].
This is the exponential-tail estimate for a compact score. All fixed
@@ -803,8 +613,8 @@ constants, not on R,L,S. Then ((eq:pre:2-2) and
(eq:pre:2-1))
|E_c X_c(F)|&le P(B)N_c(F),
- |mathcal H_c(F,G)|calH_c(F,G)|le P(B)N_c(F)N_c(G),
- |mathcal H_c(F,G)-calH_c(F,G)-barbar F_cF_c H_c G_c|
+ |H_c(F,G)|calH_c(F,G)|le P(B)N_c(F)N_c(G),
+ |H_c(F,G)-calH_c(F,G)-barbar F_cF_c H_c G_c|
&le P(B)[D_c(F)N_c(G)+N_c(F)D_c(G)].
The second equation says: ``up to slowly varying corrections, the cell
@@ -818,7 +628,7 @@ and the derivative at zero satisfies the same bound.
Finally, the Hessian identity for the ring action:
- D_u^2S_g = L^{-2}sum_cmathcal H_c(acalH_c(a,d_S u, a,d_S u).
+ D_u^2S_g = L^{-2}sum_cH_c(a,d_S u, a,d_S u).
## Stage A, part 4: augmenting with Gaussian noise
@@ -839,7 +649,7 @@ independent two-component Gaussians with covariance c_0 I, all
independent of the true ring data. Define the augmented current
J(F)=j(F)+L^{-1}sum_c
- (√(C_c),n_c+a,n_{0,c})cdot F_c.
+ (√(C_c),n_c+a,n_{0,c})· F_c.
For every constant real matrix A,
@@ -851,7 +661,7 @@ current carries the fluctuation, and we want to show its variance is at
least as big as log L/pi.
Choose a smooth trigonometric cutoff psi that is zero on the
-boundary and has band O(m), and let f_i=psivarphi_i where varphi_i
+boundary and has band O(m), and let f_i=psiphi_i where phi_i
are real orthonormal Fourier modes of low frequency |k|le4K, with
K+mll L. Define G_0, p, and theta at the cell anchors via the
discrete gradient Gram matrix. Substitute p into the density by an
@@ -940,14 +750,14 @@ noise drift, and non-anchor square-root derivative. Equation
Same-frequency Ward bound.
For the uncentred defect
[
- mathcal D(F,G)=vL^calD(F,G)=vL^{-2}sum_cbarbar F_cF_ccdot G_c-EJ(F)barJ(G),
+ D(F,G)=vL^calD(F,G)=vL^{-2}sum_cbarbar F_cF_c· G_c-EJ(F)barJ(G),
]
and for a compact eta of band O(K) with a fixed Fourier symbol,
apply the raw Ward identity (eq:pre:1-5) before replacing the
gradient by its leading symbol. The contact term cancels the raw defect;
the bracket term gives a C^1-norm bound P(B)/|k|. Hence
- |mathcal D(d_ScalD(d_Seta,G)|le P(B)/|k|+B^{-N}.
+ |D(d_S,G)|le P(B)/|k|+B^{-N}.
Applying the Ward identity earlier would have lost the exact cancellation;
this is the critical sequencing point.
@@ -956,7 +766,7 @@ this is the critical sequencing point.
Close a two-colour estimate in two steps. For perpendicular colours
r,r', and for frequencies 2Mle|k_p|le4M, put
-b_p=|k_p|^{-2} and G_p^r=r,d_S(psitildevarphi_p/|k_p|). The
+b_p=|k_p|^{-2} and G_p^r=r,d_S(psitildephi_p/|k_p|). The
Ward bound (eq:pre:5-14) applied to the two annuli in the two
colours produces a closed inequality
[
@@ -984,7 +794,7 @@ exact gradient factor in spatial direction mu is
+frac{Scos(k_{pmu}/(2S))}{i|k_p|}
[psi(z+e_mu/(2S))-psi(z-e_mu/(2S))].
]
-Under k_pmapsto-k_p the first term changes sign and the second does
+Under k_p|->-k_p the first term changes sign and the second does
not. Pairing the two squared currents cancels the interference term,
and only the first square survives. Coordinate reflections of k_p
then cancel mixed spatial terms, and the annular sum of each diagonal
@@ -993,7 +803,7 @@ has Fourier error P(B)S^{-2}, which the crude moment bound makes
negligible after summation. This yields the weighted auxiliary estimate
W^r:=sum_{|n|le4K_1}(M+|n|)^{-2}sum_{mu=1}^2
- E |J(r e_mupsi^2 e^{incdot z})|^2
+ E |J(r e_mupsi^2 e^{in· z})|^2
le Cv(v+T_w).
Notice the boxed estimate is not assumed in any of the previous steps.
@@ -1146,9 +956,9 @@ summing the opposite-sign bounds gives the upper bound.)
The O(4) plane-rotator form of Ginibre's inequality
([cite]) applies here. Two-replica substitution
-theta=varphi+psi,theta'=varphi-psi is an onto homomorphism of
+theta=phi+psi,theta'=phi-psi is an onto homomorphism of
the product tori that pushes product Haar to product Haar; expanded in
-varphi,psi, each term is a difference of sines with non-negative
+phi,psi, each term is a difference of sines with non-negative
coefficient. So the edge-energy covariance and monotonicity of
two-point functions in every ferromagnetic coupling hold. This gives the
covariance bound for the sign sectors and for the angle sector
@@ -1197,7 +1007,7 @@ reflection-positive Hilbert space. We sketch the passage, then turn to
Delta_beta.
Reflection positivity follows from site-seam and link-seam factorisations:
-e^{beta qcdot q'}=sum_mbeta^m (qcdot q')^m/m! is a sum of positive
+e^{beta q· q'}=sum_mbeta^m (q· q')^m/m! is a sum of positive
tensor products. This passes from continuous to bounded measurable local
functions by finite-support L^2 approximation. Then
inner{F}{F}=E[(Θ F)F]ge0 on bounded local functions, the
@@ -1217,7 +1027,7 @@ stated.
## Stage D, part 3: Delta_beta bound and the proof of the theorem
-Let e_t(n)=E_{t;n,n}(q_0cdot q_{e_1}) and let b be the maximum
+Let e_t(n)=E_{t;n,n}(q_0· q_{e_1}) and let b be the maximum
coupling. Use (eq:pre:11-4) with the actual maximum coupling, not
a varying threshold. For even nge Cs(b), choose the same free square
of radius comparable to n/4 around this bond in each of the n- and
@@ -1358,15 +1168,15 @@ whose aliases are bounded by C_JL^{-1}< l>^{-J}, and whose
rows, columns, and fine-difference bounds depend only on L. One can
find a positive scalar B_2 with an inequality
[
- widetilde B'U_*widetilde B'=mathsf c'^*B_2sfc'^*B_2mathsf c'sfc'
+ widetilde B'U_*widetilde B'=c'^*B_2sfc'^*B_2c'sfc'
]
such that the Schur complement U_* is nonnegative; consequently there
exists an ambient-field identity
- U_*+mathcal M^*calM^*mathcal M+calM+mathcal N^*calN^*mathcal N=I,calN=I,
+ U_*+M^*calM^*M+calM+N^*calN^*N=I,
-with mathcal M,calM,mathcal NcalN defined in terms of T,B',mathsf A_ssfA_s
-and S'=mathsf A'(sfA'(widetilde B')^{-1}(mathsf A')^*sfA')^* an orthogonal
+with M,N defined in terms of T,B',A_s
+and S'=A'(sfA'(widetilde B')^{-1}(A')^*sfA')^* an orthogonal
projection.
@@ -1379,37 +1189,37 @@ its aliases have bounds C_sL^{-1}< l>^{-s} for every
required fixed s, with R_0 vanishing through first order at p=0.
Apply the analytic alias-division lemma
(Lemma~app:alias-division) to produce
-R_l=mathsf c_l^*sfc_l^*gamma_l and H_ld'=gamma_l with
+R_l=c_l^*sfc_l^*gamma_l and H_ld'=gamma_l with
|gamma_l|+|H_l|le C_s< l>^{-s}. Set
-T_l=T_{0,l}+mathsf c_l^*H_l(B')^sfc_l^*H_l(B')^{-1}, proving
+T_l=T_{0,l}+c_l^*H_l(B')^sfc_l^*H_l(B')^{-1}, proving
(eq:free:Tidentities) with the same alias bounds. Fourier
inversion gives rows, columns, and fine differences.
For positivity, add a scalar correction to widetilde B^{-1}:
widetilde B^{-1}=B^{-1}
- +lambda B^{-1}mathsf c^*sfc^*mathsf csfc B^{-1}
+ +lambda B^{-1}c^*sfc^*csfc B^{-1}
at B and at B'. This leaves widetilde Bd=Bd, hence K, unchanged.
Choose lambdage2(C+1) and then L large enough that
widetilde B'U_*widetilde B' is positive and, after removing the
principal positivity, becomes
-mathsf c'^*B_2sfc'^*B_2mathsf c'sfc' with B_2 a positive analytic scalar.
+c'^*B_2sfc'^*B_2c'sfc' with B_2 a positive analytic scalar.
Positivity extends through p=0. Choose c_0 below the common lower
bound of widetilde B and set
ell=(widetilde B-c_0I)^{1/2}. Taking square roots: all remain
uniformly analytic on a smaller strip. Define
- S'&=mathsf A'(sfA'(widetilde B')^{-1}(mathsf A')^*,sfA')^*,
- mathcal Mz&=(calMz&=(mathsf AsfAwidetilde B^{-1}T(mathsf A')^*z,sfA')^*z,
- a^{-1/2}d'^*(mathsf A')^*z),sfA')^*z),
- mathcal Nz&=((I-S')z,calNz&=((I-S')z,
- B_2^{1/2}mathsf c'(sfc'(widetilde B')^{-1}(mathsf A')^*z).sfA')^*z).
+ S'&=A'(sfA'(widetilde B')^{-1}(A')^*,sfA')^*,
+ Mz&=(calMz&=(A B^{-1}T(A')^*z,sfA')^*z,
+ a^{-1/2}d'^*(A')^*z),sfA')^*z),
+ Nz&=((I-S')z,calNz&=((I-S')z,
+ B_2^{1/2}c'(sfc'(widetilde B')^{-1}(A')^*z).sfA')^*z).
Substitution of (eq:free:Tidentities) proves
-(eq:free:ambient) and the cokernel property; mathsf c'd'=0sfc'd'=0
-proves that mathcal N'scalN's second factor has the requested form. Using
-the definition of U_* gives mathcal M^*calM^*mathcal M+calM+mathcal N^*calN^*mathcal N=I.calN=I.
+(eq:free:ambient) and the cokernel property; c'd'=0sfc'd'=0
+proves that N'scalN's second factor has the requested form. Using
+the definition of U_* gives M^*calM^*M+calM+N^*calN^*N=I.
The extra factors are uniformly summable convolution kernels, so they
preserve the proved bounds for T. At depth zero take
widetilde B=I instead of the fine curl correction; this deletes a
@@ -1501,7 +1311,7 @@ Fisher-lite kernel that will be the "observation": for a coarse spin
V and a block configuration {U_1,...,U_{L^2}}subset S^3 with
normalised mean barbar u,u, put
- Q_b(Vmid u)= z_{ab}^{-1}exp!(ab,Vcdot barbar u),u),
+ Q_b(Vmid u)= z_{ab}^{-1}exp!(ab,V· barbar u),u),
with a,L^{-2}le ale 1, b>0 a known coupling, z_{ab} its
normalisation. This is an O(4)-symmetric positive kernel; the block
@@ -1511,7 +1321,7 @@ all block interiors, the V_b are independent.
noindentKinetic energy.
Given a fine configuration q=(q_x)_{x}, define
- mathcal E_calE_{h,t}(q) = sum_e S_t(r_e)
+ E_{h,t}(q) = sum_e S_t(r_e)
+ tfrac12sum_e m_e(q),|(ell_h dq)_e|^2,
with r_e=|d_eq|, d_eq = q_y-q_x for e=xto y, and where
@@ -1528,14 +1338,14 @@ on, the chord |(ell_hdq)_e| is at most Ct. The kernels ell_h are
the exponentially-localised rows of Lemma~lem:free-stack; each in
turn depends only on fixed L.
-The key positivity fact for the whole construction is: mathcal E_calE_{h,t}
+The key positivity fact for the whole construction is: E_{h,t}
is a sum of non-negative terms (because m_ein{0,1}). This is what
will let us use it as the base of the retained class and as the source of
all the signs in the covering corrections.
noindentObservation and the microscopic model.
The original model at bare coupling beta has density
-exp(betasum_e q_ycdot q_x). We will treat this as the scale-N
+exp(betasum_e q_y· q_x). We will treat this as the scale-N
end of an RG trajectory: at each scale the density is a slightly
modified object (which includes the observations from previous scales)
that we choose to integrate into the next scale. The observations
@@ -1568,12 +1378,12 @@ Fix once and for all:
Fix the parameters above and give the scale-j lattice
Λ_j=(Z/L^jZ)^2 on the torus of side L^j. A density
rho on ambient configurations Q=(q_x)_{xinΛ_j} is in the
-scale-j class mathscr C_j=scrC_j=mathscr C_j(b_j,t_j,scrC_j(b_j,t_j,delta_j,h)
+scale-j class C_j= C_j(b_j,t_j,scrC_j(b_j,t_j,delta_j,h)
if:
[label=(roman*)]
It has the form
- rho(q)=e^{vm^2}expBigl{-b_jmathcal E_calE_{h,t_j}(q)
+ rho(q)=e^{vm^2}expBigl{-b_jE_{h,t_j}(q)
-sum_{slots}mathbf1_Xbf1_X P_X(q)
-sum_Xmathbf1_Xbf1_X f_X(q)Bigr}Xi(q),
@@ -1627,7 +1437,7 @@ contraction.
## (c) The exact integration
-The exact integration of a class-mathscr C_jscrC_j density to
+The exact integration of a class-C_j density to
scale j-1 proceeds in three exact steps.
noindentStep 1: block observation.
@@ -1640,8 +1450,8 @@ noindentStep 2: exact MAP substitution.
The exact dependence of the kinetic energy on V is reorganised by
writing the kinetic part as
- b_jmathcal E_calE_{h,t_j}(q)
- = b_jmathcal E_calE_{h,t_j}(q)-tfrac12 a,b_j|V-Qu|^2
+ b_jE_{h,t_j}(q)
+ = b_jE_{h,t_j}(q)-tfrac12 a,b_j|V-Qu|^2
+tfrac12 a,b_j|V-Qu|^2,
i.e. we insert and remove the V-dependent Gaussian term. Combined
@@ -1692,11 +1502,11 @@ Fix the parameters L,P_0 large and H large enough so that the
hypotheses of Appendix~app:finite-choices hold. There are
H_j-dependent constants C_L,c_L>0, and there is a scale-dependent
data (b_j,t_j,delta_j,h_j) such that, for every
-j-level density in mathscr C_j(b_j,t_j,scrC_j(b_j,t_j,delta_j,h_j) and every
+j-level density in C_j(b_j,t_j,scrC_j(b_j,t_j,delta_j,h_j) and every
period m=L^j large enough, the exact integration to scale j-1 gives
a density in
[
- mathscr C_scrC_{j-1}(b_{j-1},t_{j-1},delta_{j-1},h_{j-1})
+ C_{j-1}(b_{j-1},t_{j-1},delta_{j-1},h_{j-1})
]
with
@@ -1722,56 +1532,6 @@ bounded below by a positive constant fixed once H_0 is fixed and the
depth N varies. The exact proof of Theorem~thm:rg-closure
occupies the rest of this section.
-## Proof structure
-
-The proof of Theorem~thm:rg-closure has five layers.
-
-noindentLayer 1: Schur and detection.
-Use the free-energy Schur identity of Lemma~free:bounds to split
-the exact density into the free kernel part plus a positive fluctuation
-piece. Perform the observation exactly and record the block mean
-residual (eq:rg:residual) together with its derivatives up to
-k=8.
-
-noindentLayer 2: perturbation and the Gaussian-ratio identity.
-Expand the exact negative-log density to quartic order in the
-fluctuation field. Identify the coefficients of the quadratic, cubic
-and quartic terms by a stable Taylor contraction in the free-kernel
-exponential norm. Match every coefficient against the canonical two-loop
-expansion (eq:rg:kicks). The remainder is put in the exceptional
-and error lists.
-
-noindentLayer 3: signed error bookkeeping.
-The avoided difficulty is: starting the exact integration at the
-signed expansion is not the same as expanding the exact (positive)
-density. But the exact density is positive, so its negative logarithm
-is real. We retain the exact quadratic form (bounded by the free
-kernel), and register each extra contribution as a regular error
-f_X satisfying the cap (eq:rg:error-norm). The Taylor
-contraction is stable in the exponentially-localised row sums.
-
-noindentLayer 4: covering corrections.
-All Taylor terms of degree ge5, and all the physical predicates
-(bad-edge masks, cutoff support) that would otherwise be invisible in a
-signed expansion, are reorganised as a mandatory covering gas Xi
-with bounded weight. The gas is a strict positive correction to the
-density, so it can be summed by the covering argument of
-Appendix~app:joint-majorants (in preparation, but the mechanism
-is the same as a standard polymer-gas argument).
-
-noindentLayer 5: regrouping and contraction.
-Regroup the gas into connected components; the exponential weight of
-each component is paid by the exponential localisation of ell_h,
-the classical direct cost of a bad bond, and the exceptional masses
-provided by the canonical coefficients. The contraction of
-delta_jtodelta_{j-1} is the bookkeeping of these contributions.
-
-The full analysis of each layer is in
-Sections~sec:rg-proof--sec:rg-canonical; the analytic
-inputs are in Appendices~app:analytic-block--app:joint-majorants;
-the geometric bookkeeping in Appendix~app:rg-geometry.
-
-%---------------------------------------------------------------------
# Proof of the closure theorem
%---------------------------------------------------------------------
@@ -1792,14 +1552,14 @@ constants are fixed.
## Lemma 1: The Schur observation split
[Schur observation]
-Let rhoinmathscr C_j(b,t,scrC_j(b,t,delta,h) and let Q(Vmid u) be the
+Let rhoinC_j(b,t,scrC_j(b,t,delta,h) and let Q(Vmid u) be the
positive observation kernel (eq:obs-kernel). Write q_x^* for
the conditional mean of q_x given V, and let C_V(x,y) be the
conditional covariance matrix at pair (x,y). Then the exact
integration over the fine spins decomposes as
int rho(q),Q(Vmid q),dd q
- = e^{A_0(V)}cdot R(V),
+ = e^{A_0(V)}· R(V),
where
A_0(V) is a quadratic-plus-linear expression in V
@@ -1845,11 +1605,11 @@ Let F(q) be the negative-log of the class-j density
(eq:rg:density) expanded in the fluctuation field
xi_x=q_xbarbar u_x^u_x^{-1} around the block centre barbar u_xu_x of the
coarse cell. Write
-F(xi) = F_0 + F_1cdotxi + frac12xi^TF_2xi
+F(xi) = F_0 + F_1·xi + frac12xi^TF_2xi
+frac16 F_3(xi,xi,xi) +frac1{24}F_4(xi,xi,xi,xi)
-+ mathcal R_5(calR_5(xi)
++ R_5(
for the Taylor expansion to quartic order at zero with remainder
-mathcal R_5.calR_5. Under the small-set residual bound
+R_5. Under the small-set residual bound
(eq:rg:residual) for some sufficiently large M, the coefficients
satisfy:
F_1 is O(C_LMt), and its derivatives to order k=8 are
@@ -1858,8 +1618,8 @@ satisfy:
chart;
F_3 has a bound O(C_LM^D t);
F_4 is O(C_LM^Dt^0) (bounded independently of t and L);
- mathcal R_5calR_5 has the exponential-small bound
- |mathcal R_5|calR_5|le C_Lg^5poly(M,p) in the
+ R_5 has the exponential-small bound
+ |R_5|calR_5|le C_Lg^5poly(M,p) in the
free-kernel exponential norm.
@@ -1880,9 +1640,9 @@ observation provides it; F_1,F_3 are controlled by the residual
(eq:rg:residual); F_4 is a polynomial coefficient of bounded
degree.
-The remainder mathcal R_5calR_5 is bounded by direct use of the fourth-order
+The remainder R_5 is bounded by direct use of the fourth-order
remainder formula
-mathcal R_5(calR_5(xi)=int_0^1((1-s)^4)/(4!)F^{(5)}(sxi)(xi^{ ⊗ 5})ds,
+R_5(-s)^4)/(4!)F^{(5)}(sxi)(xi^{ ⊗ 5})ds,
with the free-kernel exponential-sum bound controlling F^{(5)}.
Since F_2 is roughly the free kinetic operator with Hessian c_0,
the free-kernel exponential-sum bound for its inverse (which is our
@@ -1898,7 +1658,7 @@ Let f_X be a regular error slot of the class-j density with cap
delta (in the norm (eq:rg:error-norm)). Under one exact RG step,
its image through the Taylor map plus the Schur split satisfies
- |mathcal TcalT f_X|_{k,j-1,A}
+ |T f_X|_{k,j-1,A}
le (C/L+o(1)),delta,.
More generally, if A_j denotes the error coefficient in the class at
@@ -1910,8 +1670,8 @@ with H_j the scale-j coupling.
Proof.
-The composition operator mathcal TcalT is bounded on the norm
-|cdot|_{k,j,A} by a constant q_c<1 provided L is chosen large
+The composition operator T is bounded on the norm
+|·|_{k,j,A} by a constant q_c<1 provided L is chosen large
enough. The bound comes from two sources:
the free-kernel row sum bound
(eq:free:derivative) contributes a factor C/L because
@@ -1975,12 +1735,12 @@ For reflection positivity: the fine reflection at a retained seam
exchanges the two halves of the block partition. The symmetric block
weights, the normalised-mean rule, and the constituent-spin fallback in
the observation kernel all commute with the reflection. For a function
-F of output spins on one side, let mathcal QFcalQF be its conditional
+F of output spins on one side, let QF be its conditional
expectation given the input spins on that side. Conditional independence
and reflection covariance give
[
mu_{rm out}((thetaoverlineoverline F)F)F)F)
- =mu_{rm in}((thetamathcal QFbar)calQFbar)mathcal QF)calQF)ge0.
+ =mu_{rm in}((thetaQFbar) QF)calQF)ge0.
]
Thus reflection positivity passes through each blocking step. Translation
and reflection compatibility on the retained lattice pass through the
@@ -1991,8 +1751,8 @@ covariance; that is the whole content.
## Lemma 6: Assembly
[Assembly]
-Let rhoinmathscr C_j(b,t,scrC_j(b,t,delta,h). Then under one exact RG step,
-rho is mapped to rho'inmathscr C_scrC_{j-1}(b',t',delta',h') with
+Let rhoinC_j(b,t,scrC_j(b,t,delta,h). Then under one exact RG step,
+rho is mapped to rho'inC_{j-1}(b',t',delta',h') with
b'-b = alpha+kappa/b+..., delta' le H_j^{-1/20}delta,
@@ -2007,9 +1767,9 @@ Lemma~lem:rg-cover-regroup to control the higher-order remainder
and the physical masks, then Lemma~lem:rg-symmetry to transport
symmetry and reflection positivity. The new coupling b' is read from
the coefficient of the quadratic kinetic term in the output:
-b'=b+F_2^{(2)}/mathcal E_calE_{h,t}^{(2)}, where F_2^{(2)} is the
+b'=b+F_2^{(2)}/E_{h,t}^{(2)}, where F_2^{(2)} is the
quadratic part of the exact negative logarithm and
-mathcal E_calE_{h,t}^{(2)} is the quadratic kinetic term. Matching
+E_{h,t}^{(2)} is the quadratic kinetic term. Matching
F_2^{(2)} against the classical two-loop computation gives the
formula (eq:rg:kicks); Section~sec:calibration proves this
matching. The new cutoff t' and the new error cap delta' are
@@ -2047,7 +1807,7 @@ axis and write
sum_xu_x=0, z_x=sqrtbeta,u_x.
Under the nearest-neighbor measure ddmu_beta the sum
-sum_e q_ycdot q_x becomes
+sum_e q_y· q_x becomes
sum_e[1-tfrac12|u_y-u_x|^2+O(|u|^3)], so the quadratic fluctuation
is a massive Gaussian of mass sim nearest-neighbor Laplacian, and the
cubic and higher terms are suppressed by 1/sqrtbeta per spin.
@@ -2190,7 +1950,7 @@ Choose N such that beta_N=beta, the desired bare inverse coupling.
Then to leading order,
N=(beta-H)/(gamma)-(1)/(2pigamma)sum_{h=0}^{N-1}(1)/(beta_h)+O(1)
- =(beta)/(gamma)-frac12(logbeta)/(gamma)cdot(1)/(pi)+O(1),
+ =(beta)/(gamma)-frac12(logbeta)/(gamma)·(1)/(pi)+O(1),
using the Euler-Maclaurin asymptotics sum_{h=0}^{N-1}1/beta_h
sim(gamma)/(gamma^2)int ds/beta(s)