% low_temp_sk_glauber_typical_v2.txt % A typical-start upper bound for low-temperature SK Glauber dynamics. % Single-file clean rewrite. Read top to bottom. usetikzlibrary{arrows.meta,calc,positioning} allowdisplaybreaks # A typical-start upper bound for low-temperatureSherrington--Kirkpatrick Glauber dynamics maketitle Abstract: We study single-site heat-bath (Glauber) dynamics for the zero-field Sherrington--Kirkpatrick spin glass at a fixed inverse temperature beta>1. We prove that a configuration sampled from equilibrium mixes to equilibrium in subexponential time: for every fixed beta>1 there is a constant c_beta>0 such that, with probability tending to one over the disorder and the starting configuration, the total-variation distance from equilibrium is below 1/4 by time exp(n^{1-c_beta}). The proof admits the explicit value c_beta=1/40000000 for all fixed beta>1. The starting configuration is held fixed before the distance to equilibrium is measured; this is the essential subtlety, since averaging the start into the law would make the distance zero at every time. tableofcontents %====================================================================== # Introduction %====================================================================== ## The question in plain terms The SK model is a mean-field spin glass: n spins sigma_iin{-1,1} interact through random pairwise couplings. The heat-bath (Glauber) dynamics is the natural local sampler. Each site has an independent rate-one clock; when the clock at site i rings, the spin sigma_i is replaced by a fresh value drawn from its conditional law given all the other spins. The chain has 2^n states and is reversible with respect to the Gibbs measure pi^J_{n,beta}. At high temperature the chain mixes quickly, in polynomial time from every start. At low temperature (beta>1 in the zero-field SK model) the landscape is glassy: there are deep valleys separated by high barriers, and a configuration placed in the wrong valley can take a very long time to escape. Sellke [cite] shows the worst starting configuration can force an exponential lower bound on the mixing time. That leaves a sharper question. An adversarial start is unrealistic for a sampler run on a freshly drawn instance. What happens if we start the chain from a typical equilibrium configuration, i.e. from one sample sigma_0simpi^J_{n,beta}? This paper answers the question for every fixed beta>1: the chain is fast, at most subexponential in n, from such a start. Two conventions are essential and we fix them now. [label=(roman*)] The start is random and then held fixed. We first draw the disorder J, then one configuration sigma_0simpi^J_{n,beta}, and only then look at the law P_t^{J,beta}(sigma_0,cdot) of the chain started at that single sigma_0. The total-variation distance in the mixing time is between this fixed-start law and pi^J_{n,beta}. If one instead averaged sigma_0 into the starting distribution, stationarity would force the distance to be zero at every time and the question would vanish. Time is continuous, one clock per site. Each site rings at rate one, so the total jump rate of the chain is exactly n. ## The model Let nge2 and let (J_{ij})_{1le i1 there is a constant c_betain(0,1) such that [ mathbb P_bbP_{J,,sigma_0simpi^J_{n,beta}}Bigl{ T^J_{n,beta}(sigma_0)leexp(n^{1-c_beta})Bigr} → 1 (nto∞). ] The constant may be taken as c_beta=1/40000000 uniformly over every fixed beta>1, and it does not depend on the realized disorder. Equivalently, in probability over the disorder J, the Gibbs mass of the set of configurations whose mixing time is at most exp(n^{1-c_beta}) tends to one. The exponent is not optimized, and the dimension from which the estimates hold may depend on the fixed beta; all such thresholds are absorbed into ``tending to one''. The companion lower bound in [cite] gives a matching stretched-exponential barrier. Combined with Theorem~thm:main it produces a two-sided statement. [Two-sided typical-start bound] For every fixed beta>1, in the model and continuous-time convention above, [ mathbb P_bbP_{J,,sigma_0simpi^J_{n,beta}}Bigl{ exp(n^{1/10000})1/4 for sigma_0simpi^J_{n,beta}. By (eq:T) and continuity of the finite-state semigroup this forces the strict lower bound T^J_{n,beta}(sigma_0)>exp(n^{1/10000}). The upper bound is Theorem~thm:main. Both events use the same joint law of (J,sigma_0) and the same rate-one clocks, so their intersection has probability tending to one. ## How the proof works The proof has five logical stages. We describe each in words, then point to its section. [label=Stage arabic*.] Equilibrium scalar inputs. The Parisi theory of the SK model provides a scalar potential Phi and a magnetization martingale whose variance in overlap space is dictated by the minimizing Parisi measure. We state exactly which equilibrium inputs we use, in the normalization of this paper. See Section~sec:scalar. The long diffusion and its entropy--energy identity. We continue the scalar diffusion past the endpoint of the Parisi support, where the magnetizations pin down (level sets shrink exponentially). An exact It^o calculation produces a scalar identity, equation~(eq:scalar-balance), that balances the information acquired by matrix queries against the energy gained. This is the engine of the entropy bound. See Section~sec:scalar. The Gaussian-query proposal. We build a random spin by querying the matrix W=W(G) along orthogonalized directions. Each query is conditionally Gaussian given the past and constructs a fresh coordinate of a driving Brownian motion. The proposal Q_l has two controlled features: sublinear relative entropy {rm KL}(Q_lVert P), and a likelihood bound outside an exponentially unlikely event. See Section~sec:proposal. Reversing the entropy: coverage. Forward entropy alone says nothing about reaching a prescribed set. Conditioning the Gibbs joint law on the spin makes the remaining disorder law strongly log-concave; a small Gaussian smoothing plus the quadratic transport inequality then reverses the entropy comparison. The outcome is a coverage statement: any Gibbs set of mass at least n^{-d} receives at least exp(-n^{1-2d}) proposal mass for the realized target disorder and smoothing base. See Section~sec:reverse. Balancing flows and mixing. Two independent proposals joined across a cut contradict the avoidance of a small boundary unless every cut of appreciable Gibbs mass carries stationary flow at least B_{min}=exp(-n^{1-b/20}). The geometry is elementary: a short sequence of spins crossing a cut must contain a spin close to the boundary. Fine-horizon spins supply endpoint coverage; coarse-horizon spins have rare enough failures along the connecting sequence, and rotating the Gaussian inputs makes consecutive coarse spins close in Hamming distance. Then a finite-state pruning lemma and entropy dissipation turn the flow bound into a statement about typical fixed starting states. See Sections~sec:connections and~sec:mixing. Theorems~thm:main and cor:two-sided are assembled from these stages at the end of Section~sec:mixing. ## Relation to prior work High-temperature, worst-start. Eldan, Koehler and Zeitouni cite[Section~6]{EKZ2022} proved polynomial worst-start mixing for beta<1/4. Wang cite[Theorem~1.1]{Wang2026} established optimal-order Glauber mixing for every fixed beta<1/2, and Boban, Li and Oveis Gharan cite[Corollary~1.6]{BLO2026} extended polynomial worst-start mixing in zero field to beta<1/2+ε_0 for some absolute ε_0>0. Low-temperature, worst-start. Sellke cite[Theorem~1.1]{Sellke2025} proved exponential worst-start lower bounds and distinguished the worst-start obstruction from fast mixing after Gibbs-distributed start. Theorem~thm:main resolves the latter throughout the fixed low-temperature phase. Equilibrium Parisi theory. The stationarity conditions that drive the argument come from the variational theory of the Parisi functional: Chen cite[Theorem~2, Proposition~1]{Chen2017}, Jagannath and Tobasco cite[Proposition~1.1, Corollaries~3.6,~3.10]{JT2017}, and Zhou cite[Theorem~1]{Zhou2026}. For the whole low-temperature phase we use full interval support from Lopatto cite[Theorem~1.1]{Lopatto2026} (version~3). Guerra cite[Theorem~3]{Guerra2003} supplies the pressure upper bound, and Auffinger--Chen cite[Propositions~1--2]{AC2015} supplies spatial regularity and step-approximation stability of the Parisi equation. Techniques. The Gaussian-query construction is the exact finite-dimensional analogue of Bolthausen's iterative TAP construction cite[Sections~2--3]{Bolthausen2014} and of Montanari's incremental approximate message passing cite[Sections~2--3]{Montanari2021}: query, orthogonalize, reveal. The reverse-entropy step uses triangular transport under strong convexity cite[Corollary~3.10]{BKM2005} together with the convolution comparison recorded in cite[Lemma~11]{LugosiNeu2022}. The mixing step combines the coarea proof of the reversible Cheeger inequality cite[Theorem~3.5]{LawlerSokal1988} with entropy dissipation for finite Markov chains cite[Lemmas~2.5,~2.7]{DiaconisSaloffCoste1996}. %====================================================================== # Notation and Gaussian completion %====================================================================== Fix a finite beta>1. Constants may depend on this fixed parameter; when that dependence matters we write C_beta or O_beta(cdot). All logarithms are natural. Distances and inner products.: For probability measures mullnu on a common space, set {rm KL}(muVertnu)=intlog(dmu/dnu),dmu. On the spin cube, total variation is one half of the ell^1 distance. For vectors v,winmathbb R^nbbR^n write [ < v,w>_n=frac1nsum_i v_i w_i, |v|_n^2=< v,v>_n . ] The symbols |v| and |A| denote the Euclidean norm of a vector and the operator norm of a matrix. The GOE completion.: A heat-bath update at site i never sees J_{ii}. We complete the interaction matrix by adding independent diagonal entries of law N(0,2/n): define a symmetric matrix W=W(G) whose entries above the diagonal are W_{ij}=J_{ij}/sqrt n and whose diagonal entries are independent N(0,2/n). Write G for the vector of independent standard Gaussian drivers of all these entries and gamma for its law. Since x_i^2=1 for every spin, [ tfrac12,x^{mathsf T}sfTdiag(W),x =tfrac12sum_i W_{ii} ] is the same random constant for every configuration. Hence the diagonal contributes the same additive constant to every Gibbs weight; it changes neither pi^J_{n,beta} nor any heat-bath rate nor the transition kernel. Its effect on mathbb EbbE F(G) is a deterministic constant that we absorb into the pressure functional mathcal PcalP below. Gibbs weights on the completed matrix.: Define, with W=W(G), [ widehatwidehat H_G(x)=H_G(x)=tfrac12 x^{mathsf T}sfTW(G),x, F(G)=logsum_{xin{-1,1}^n}e^{betawidehatwidehat H_G(x)}H_G(x), pi_G(x)=e^{betawidehatwidehat H_G(x)-F(G)}H_G(x)-F(G). ] Because the diagonal is a configuration-independent constant, pi_G=pi^J_{n,beta} as probability measures. We work with widehatwidehat H_GH_G everywhere. The Gibbs joint law of disorder and spin is [ P(dG,x)=gamma(dG),pi_G(dx). ] This is exactly the joint law of the starting configuration in Theorem~thm:main. It is crucial that P is a joint law: downstream we compare a proposal law Q_l to P on the joint (G,x)-space. The transition kernel for a fixed disorder and a fixed initial configuration is a different object, and the final mixing argument must separate the two carefully. %====================================================================== # Scalar inputs and the entropy--energy identity %====================================================================== This section extracts the one-dimensional input that drives the whole proof. The endpoint is Proposition~lem:scalar-balance (equation~(eq:scalar-balance)): an exact scalar balance between the energy a matrix-query proposal gains and the information it pays for. Everything here is about a single diffusion and a few deterministic functions of a scalar time; no high-dimensional geometry enters yet. ## Equilibrium inputs, stated once Let mu_beta be a minimizing Parisi probability measure for the zero-field Gaussian SK model at the fixed finite beta>1. We use the following external inputs. [leftmargin=*] Pressure. Guerra's finite-volume interpolation bound cite[Theorem~3]{Guerra2003}: mathbb EbbE F(G)le nmathcal P,calP, where mathcal PcalP is the Parisi pressure functional, whose exact normalization we give below. Regularity and stability. Auffinger--Chen cite[Propositions~1--2]{AC2015}: the scalar Parisi equation has continuous spatial derivatives, and its solution is stable under step approximation of the CDF of mu_beta. Full support and stationarity. Lopatto cite[Theorem~1.1, Lemma~2.2, Proposition~2.4, Lemma~2.5]{Lopatto2026}: the minimizing measure has full interval support, [ supp(mu_beta)=[0,q_*], 0v_* we continue the potential by the explicit formula Phi(v,y)=log(2cosh y)+(beta^2-v)/(2), alpha_v=1 . This agrees with the original solution up to beta^2 and satisfies the same PDE for every vge v_*. It describes the regime in which the magnetizations pin down: the level sets of the magnetization shrink and the spin becomes deterministic. Continue for every vge0 the two scalar functions [ m(v,y)=Phi_y(v,y), a(v,y)=Phi_{yy}(v,y), ] so that on the extension m(v,y)=tanh y and a(v,y)=1-tanh^2 y. Let Z be a standard Brownian motion and let Y solve dY_v=alpha_v,m(v,Y_v),dv+dZ_v, Y_0=0 . The drift is bounded and spatially Lipschitz, so the scalar SDE has a unique strong solution. Abbreviate m_v=m(v,Y_v) and a_v=a(v,Y_v). The stationarity identity, the differentiated PDE, and It^o's formula give the key scalar facts m_0=0, dm_v=a_v,dZ_v, mathbb EbbE m_v^2=(v)/(beta^2)(0le vle v_*), and this martingale equation continues past v_* by direct It^o calculus with the explicit functions (eq:phi-extension). We use, repeatedly, the deterministic bounds 00 maps a profile f to [ (Tf)(y)=frac1lambdalogmathbb EbbE, e^{lambda f(y+sqrtdelta U)}, Usim N(0,1), ] with Tf(y)=mathbb EbbE f(y+sqrtdelta U) when lambda=0. Write mathbb E_bbE_lambda for the Gaussian expectation tilted by e^{lambda f(y+sqrtdelta U)} and renormalized. Differentiating gives the exact formulas [ (Tf)'=mathbb E_bbE_lambda f', (Tf)''=mathbb E_bbE_lambda f''+lambdaVar_lambda(f') . ] Suppose 0v_*, the density of Y_v divided by cosh y is symmetric and nonincreasing in |y|. It suffices to check this for step CDFs and pass to the limit; stability cite[Propositions~1--2]{AC2015} justifies the passage. For a step CDF with masses alpha^{(j)} on intervals [v_j,v_{j+1}], and varphi_t the centered Gaussian density of variance t, the transition density on one interval is [ K_j(z,y)=varphi_{v_{j+1}-v_j}(y-z) expbigl{alpha^{(j)}[Phi(v_{j+1},y)-Phi(v_j,z)]bigr} . ] The Gaussian recursion normalizes K_j; its Doob transform has drift alpha^{(j)}Phi_y. In a product of kernels the intermediate factors at y_j reduce to [ expbigl{-(alpha^{(j)}-alpha^{(j-1)})Phi(v_j,y_j)bigr}, ] which is even and nonincreasing in |y_j| because the masses are nondecreasing and Phi is even and convex. The first positive-time Gaussian convolution turns the initial point mass at zero into a symmetric decreasing density, and multiplication by these factors plus further Gaussian convolutions preserve that property (using the centered-interval level-set representation of a symmetric decreasing profile). Extending the final mass to one leaves an endpoint factor proportional to cosh y; dividing by it leaves the asserted symmetric decreasing profile. Weighting either sechy or sech^3y by a decreasing profile can only increase the expectation of sech^2y, since two nonincreasing functions of |y| have nonnegative covariance. Integration by parts gives [ frac{int_{mathbb R}bbRsech^{k+2}y,dy} {int_{mathbb R}bbRsech^{k}y,dy} =(k)/(k+1)(k>0). ] Consequently r/pge1/2 and mathbb EbbE a_v^3/r(v)ge3/4. Differential identities. On the extension a_v=1-m_v^2, and direct It^o calculations yield [ p'=-r, r'=4r-6,mathbb EbbE a_v^3 . ] The two moment ratios imply p'le-p/2 and r'le0. Since 0v_* define the two scalars that will be the energy per spin and the information cost per spin of the matrix proposal: e(V)=int_0^V√(r(v)),p(v),rho(v),dv, j(V)=frac12int_0^Vrho(v)^2, mathbb E(a_v^2)/(1-m_v^2)bbE(a_v^2)/(1-m_v^2),dv . The factor rho(v)^2 inside j accounts for the independent noise we introduce in the proposal. Below the support endpoint, It^o's formula together with the PDE and stationarity gives [ mathbb E[Y_bbE[Y_{v_*}m_{v_*}]=int_0^{v_*}(p(v)+alpha_v v/beta^2),dv, mathbb EbbEPhi(v_*,Y_{v_*})=Phi(0,0) +frac12int_0^{v_*}alpha_v v/beta^2,dv . ] These also follow by step approximation with bounded first and second spatial derivatives. Substituting the endpoint value (eq:phi-at-vstar) and using mathcal H(tanhcalH(tanh y)=log(2cosh y)-ytanh y gives int_0^{v_*}p(v),dv+mathbb EbbE,mathcal H(m_calH(m_{v_*}) =mathcal P-(calP-(beta^2)/(4)(1-q_*)^2 . [Entropy--energy identity] For every V>v_*, log2-j(V)+beta e(V)=mathcal P-(calP-(beta^2)/(4),p(V)^2 . Proof. The same martingale calculation applied to binary entropy yields [ frac12int_0^{v_*}mathbb E(a_v^2)/(1-m_v^2)bbE(a_v^2)/(1-m_v^2),dv =log2-mathbb EbbE,mathcal H(m_calH(m_{v_*}). ] Localization justifies It^o's formula near z=pm1; the integrand is bounded because ale1-m^2. Since rho=1 and betasqrt r=1 on [0,v_*], the two identities above prove (eq:scalar-balance) at V=v_*. Above v_* the derivative of the left side of (eq:scalar-balance) is (beta^2/2),r(v),p(v): indeed a_v=1-m_v^2 there, and rho^2=beta^2 r. The derivative of -beta^2p(v)^2/4 is the same, by p'=-r. Integrating from v_* therefore proves the claim for every V>v_*. The terminal error (beta^2)/(4)p(V)^2 on the right of (eq:scalar-balance) is at most C_beta e^{-V} by (eq:1). We take V proportional to log n, so this error is a negative power of n while the spin is nearly pinned. %====================================================================== # The Gaussian-query proposal %====================================================================== We now build a random spin using the matrix W=W(G) together with auxiliary randomness. The construction has a single design principle: each query of the matrix is orthogonal to the earlier queries and is independent given the past. This exposes fresh Gaussian information while leaving a compressed GOE block unrevealed, so that a clean conditional-Gaussian argument applies at every step. The scalar identity (eq:scalar-balance) will then control both the relative entropy of the resulting law Q_l and its likelihood outside a small exceptional event. The spirit is the same as Bolthausen's iterative TAP construction cite[Sections~2--3]{Bolthausen2014} and Montanari's incremental approximate message passing cite[Sections~2--3]{Montanari2021}, both of which reveal Gaussian information along orthogonalized directions. What is new here is the exact conditional-law bookkeeping for a finite mesh, including the independent auxiliary noises that complete each increment. ## Construction and conditional Gaussian laws Mesh.: Fix a horizon parameterfootnote{We use the letter l both for the horizon parameter and, later, for the horizon value. Context disambiguates. The final paper uses l=f for fine coverage and l=b for the coarse connecting sequence; see Section~sec:reverse.} 0v_* with v_* the support endpoint of Section~sec:scalar. The two driving streams.: The disorder G retains its ordinary Gaussian law gamma. We build a vector-valued process whose coordinates follow the scalar SDE (eq:Y), driven by two independent streams, dZ=rho(v),dB+√(1-rho(v)^2),dD . Here D is an auxiliary standard Brownian motion, independent of everything; B is to be constructed from matrix queries together with auxiliary completions so that, after averaging over the disorder, (B,D) is a standard Brownian pair. By (eq:1) the weight rhole1 equals one up to v_* and is nonincreasing afterwards; this is exactly what keeps the level sets pinning while retaining unit variance in Z. Write M(v)_i=m(v,Y_{v,i}). Building B.: On the first interval [0,h] we generate B entirely from independent auxiliary Brownian noise. For each subsequent interval [t_k,t_{k+1}], 1le kk}=I-Pi_{le k}. Query the column Wu_k. For an independent standard Gaussian vector z_k, set the endpoint of the next Brownian increment to be g_k=h^{-1/2}(B(t_{k+1})-B(t_k)) =Pi_{>k}Wu_k+Pi_{le k}z_k . Fill the interval by linear interpolation between its endpoints plus an independent centered Brownian bridge. Every new bridge and completion noise is fresh. A deterministic-coefficient integral against this filled path may equivalently be obtained by conditioning Brownian motion on its endpoint. The spin.: At time V, given the path, draw the coordinates x_iin{-1,1} independently with means M_i(V). Let Q_l be the marginal law of (G,x) produced this way. Why (B,D) is a standard Brownian pair.: Condition on the columns already revealed and on the auxiliary inputs already used. Gaussian orthogonal invariance leaves the unqueried block a GOE matrix on the orthogonal complement, with the original variance scale. The next direction u_k is measurable with respect to this past. In an orthonormal basis beginning with u_k/sqrt n, the new diagonal entry, the remaining entries of that column, and the next compressed block are independent. Hence Pi_{>k}Wu_k is a fresh Gaussian with covariance Pi_{>k}, and [ d_k=< u_k,Wu_k>_nsim N(0,2/n) ] conditionally, before the column is revealed. The independent completion has covariance Pi_{le k}, so g_k in (eq:gk) is a fresh standard Gaussian vector. Induction, followed by the independent bridge filling, proves that (B,D) in (eq:4) is a standard Brownian pair. Coordinate symmetry.: After forgetting the matrix, the coordinate paths and their final spins are i.i.d. The conditional mean of x given the streams through time v is M(v), and its unconditional law is uniform on the cube by the symmetry of the scalar process. The spin marginal of P is also uniform, by coordinate-sign symmetry of the disorder. Both are facts about marginals; the proposal spin remains correlated with its own matrix. ## Entropy and likelihood estimates The next proposition supplies two distinct comparisons. Entropy will yield coverage after smoothing; the likelihood estimate will keep intermediate proposals out of the small boundary of a putative low-flow cut. Both are stated for the joint law Q_l on (G,x)-space. [Proposal comparison] For every fixed 0_n, the empirical event gives [ |c-s|&le Cetalog(2K), left|(mathbf1_bf1_{jle k}< u_j,g_k>_n)_{jk}right| &le Cetalog(2K), bigl|(< g_k,x>_n)_k-bbigr|&leeta . ] The middle estimate uses that left multiplication of the population cross matrix by the lower-triangular R^{-mathsf T}sfT preserves its zero upper triangle, including the diagonal. Also |c|le1, because the query directions are orthonormal in |cdot|_n. Energy of the final spin.: Let H_0=Pi_{>K-1}. Revealing the columns in order decomposes the quadratic form exactly as [ frac{widehatwidehat H_G(x)}H_G(x)n =sum_kc_k< x,Pi_{>k}g_k>_n +tfrac12sum_kc_k^2d_k +tfrac12< x,H_0WH_0x>_n . ] The last two terms are O(eta) outside an event of probability at most CKexp(-cneta^2): the diagonal estimate follows from the conditional Gaussian law and sum c_k^2le1, while the residual term uses that the spin has not seen the unrevealed block. Removing the projections in the first term costs at most Cetalog(2K) by the triangular bound. Its remaining value is scdot b up to the same error. The bound on (log r)' in (eq:1) implies s_k/sqrt h differs from √(r(v)) on [t_k,t_{k+1}] by a relative O(h). The missing initial interval contributes O(h), and e(V) is bounded. Hence scdot b=e(V)+O_beta(h), and altogether widehatwidehat H_G(x)/n=e(V)+O_H_G(x)/n=e(V)+O_beta(h+etalog(2K)). This is the energy contribution to the comparison. We next isolate the entropy paid for the matrix information alone. The auxiliary reference law.: Start on an enlarged path space with an independent uniform spin x, the ordinary Gaussian disorder, and all unconditioned proposal inputs. Call this the unlinked experiment. To obtain the actual proposal, multiply the density of the unlinked experiment by [ prod_i(1+x_i,m(V,Y_{V,i})). ] For fixed x, the scalar martingale shows this is a stochastic exponential with coordinate integrands [ xi_i(v)=frac{x_i,a(v,Y_{v,i})}{1+x_i,m(v,Y_{v,i})}, |xi_i(v)|le2, ] against the streams Z_i in (eq:4). Call the resulting law the linked experiment. Summing the linking density over the uniform spin shows the path and disorder marginal is unchanged. For a continuous local martingale N starting at zero write mathcal E(N)_V=calE(N)_V=exp(N_V-< N>_V/2). The linking density factors as E_BE_D with [ E_B&=mathcal E(calE(sum_iint_0^{cdot}xi_irho,dB_i)_V, E_D&=mathcal E(calE(sum_iint_0^{cdot}xi_i√(1-rho^2),dD_i)_V, ] using the zero quadratic covariation of B and D. Define the auxiliary reference to have density E_D relative to the unlinked law. Its (G,x)-marginal is again the independent product of Gaussian disorder and uniform spin. The conditional mean-one assertion behind this fact needs a filtration check, since the matrix-query endpoints depend on the past of D. Filtration check.: Let mathcal AcalA contain all primitive auxiliary inputs except D, and under the unlinked law set [ mathcal F_v=calF_v=sigma(x,B_{[0,v]},D_{[0,v]}), mathcal G_v=calG_v=sigma(x,G,mathcal A,D_calA,D_{[0,v]}). ] Interval by interval, every query direction is determined by past D and by information in mathcal G_0.calG_0. Each new endpoint and its entire bridge are known before using the current D-increment. Deterministic-coefficient bridge integrals are linear Gaussian integrals with the endpoint term fixed by that past. Thus mathcal F_v⊆calF_v⊆mathcal G_v.calG_v. One can also see this pathwise: rho=1 before v_*, and (eq:1) gives -rholerho'le0 afterward, so rho has bounded variation and the identity intrho,dB=rho B-int B,drho, applied on successive intervals, uses no future D. The pair (B,D) is Brownian in mathcal F,calF, and D is Brownian in mathcal GcalG because its time-zero information is independent of D. The bounded continuous D-integrand has the same stochastic integral in these two filtrations: identical left-endpoint approximations converge in L^2 by the It^o isometry. The bounded-integrand exponential-martingale theorem in mathcal GcalG now gives conditional mean one given (x,G,mathcal A),calA), proving the claimed marginal. All later Brownian changes of measure use mathcal F;calF; no Brownian property for B in the enlarged filtration is required. Information cost and likelihood.: The density of the linked experiment relative to the auxiliary reference is E_B, whose logarithm is [ L=log E_B=sum_i(int_0^Vxi_irho,dB_i -tfrac12int_0^Vxi_i^2rho^2,dv). ] Under the linked law, compensating the B_i-drift by rhoxi_i leaves a centered integral with bracket at most 4nV, so it has exponential-martingale tails. The remaining half-bracket is a sum of independent coordinate variables, each bounded by 2V. Coordinate independence holds after linking because the linking density factors across the i.i.d. coordinate streams and uniform spins; conditionally on the streams through time v, the linked spin has probabilities (1pm m(v,Y_{v,i}))/2. We obtain mathbb EbbE L=nj(V), L=nj(V)+O(neta), outside an event of probability at most exp(-n^{1-4l}) for large n. The centered integral is controlled by exponential Markov; the half-bracket by bounded-variable concentration. The Brownian drift change follows either from Girsanov's theorem for bounded integrands, or from bounded predictable step approximation together with the mean-one exponential identities. Changing the reference to P.: Change the auxiliary reference a second time, multiplying only by the density of P relative to independent disorder and uniform spin. The new reference has (G,x)-marginal P, and the log density of the linked experiment relative to it is [ L-nlog2+F(G)-betawidehatwidehat H_G(x).H_G(x). ] Differentiating the log partition function gives a Gaussian Lipschitz constant at most C_betasqrt n. The Gaussian concentration inequality cite[Section~5.4, Theorem~5.6]{BLM2013} together with the pressure bound (eq:pressure) therefore implies [ F(G)le nmathcal P+ncalP+neta ] outside an event of probability at most 2exp(-c_beta neta^2). This uses the ordinary Gaussian marginal of G, which is unchanged after linking. Substituting the scalars.: Insert (eq:scalar-balance), (eq:7), p(V)^2le C_beta e^{-V} and (eq:L-conc) into the log density. On the events just proved it is at most [ C_beta n(h+etalog(2K)+e^{-V})=o(n^{1-l/2}). ] Indeed V=llog n+O(h), log(2K)=O(log n), and 0B_{min}:=exp(-n^{1-b/20}) quadfor every S with min(pi_G(S),pi_G(S^{rm c}))ge n^{-d}. Proof. We split the argument into (a) the geometry of a putative low-flow cut, (b) the construction of a proposal sequence, and (c) two estimates that make this sequence impossible. ## A low-flow cut has a small enlarged boundary For a sufficiently large fixed C, Gaussian quadratic-form tails on a 1/4-net give |W(G)|le C except with probability O(e^{-cn}). On this event every one-spin energy difference is O(sqrt n), because flipping a single spin changes x^{mathsf T}sfTWx/2 by an O(|W|sqrt n) amount. The heat-bath rate across any one-spin edge is therefore at least exp(-O_beta(sqrt n)). If a cut has stationary flow at most B_{min}, then its two-sided vertex boundary [ ∂ S={x:,d_H(x,S)=1 and d_H(x,S^{rm c})=1} ] has Gibbs mass at most 2B_{min}exp(O_beta(sqrt n)), since each boundary vertex controls at most 2n unit-rate crossings each carrying a heat-bath acceptance probability at least exp(-O_beta(sqrt n)). Set R_n=lceil n^{1-b/4}rceil and enlarge ∂ S by Hamming distance R_n. For two spins at Hamming distance at most R_n and with |W(G)|le C, the logarithmic Gibbs weight ratio is bounded by O_beta(n^{1-b/8}): the change in widehatwidehat H_GH_G along any path of R_n flips is at most Csqrt ncdot R_n/sqrt ncdot n^{1/2}=O_beta(n^{1-b/8}), and a Hamming ball of radius R_n has logarithmic cardinality O(n^{1-b/4}log n). The enlarged boundary mathcal D_R(G)calD_R(G) therefore has target Gibbs mass at most exp(-tfrac12 n^{1-b/20}) for n large, since 1-b/20<1-b/8 and b/200 [ sum_{yne x}c(x,y)le Rqquad(xinΩ). ] Let 0<ε<1/3 and B>0. Suppose that [ Q(S,S^{rm c}):=sum_{xin S,,ynotin S}pi(x)c(x,y)>B quadwhenever εlepi(S)le1-ε. ] Then for every T>0, sum_{xinΩ}pi(x)bigl|P_T(x,cdot)-pibigr|_{rm TV} lefrac{√(2R,H(pi))}{Bsqrt T}+2ε, H(pi)=-sum_xpi(x)logpi(x). Proof. Put q(x,y)=pi(x)c(x,y)=q(y,x), and use the unordered-edge Dirichlet form [ mathfrak D(u)=frakD(u)=sum_{{x,y}}q(x,y)(u(x)-u(y))^2. ] Only distinct vertices occur, and every unordered pair is counted once. Step 1: removing sets that do not expand.: Start with C=Ω. Whenever a nonempty Usubset C satisfies [ pi(U)lepi(C)/2, Q(U,C∖ U)1-ε and Q(U,C∖ U)ge Bpi(U) quadfor every Usubset C with pi(U)lepi(C)/2 . Step 2: a Poincar'e inequality on the retained set.: Use the coarea proof of the reversible Cheeger inequality cite[Theorem~3.5]{LawlerSokal1988}, keeping the original unnormalized weights on C. The resulting estimate is sum_{yin C}pi(y)(u(y)-barbar u_C)^2u_C)^2 le(2R)/(B^2),mathfrak D(u),frakD(u), barbar u_C=u_C=frac1{pi(C)}sum_{yin C}pi(y)u(y). To prove it, choose a weighted median m of u on C. Each of w=(u-m)_+ and w=(m-u)_+ has support of mass at most pi(C)/2. Integrating (eq:internal-expansion) over the level sets of w^2 gives [ Bsum_Cpi w^2 lesum_{{x,y}subset C}q(x,y)|w(x)^2-w(y)^2|. ] By Cauchy--Schwarz this is at most [ (sum_{{x,y}subset C}q(x,y)(w(x)-w(y))^2)^{1/2} (2Rsum_Cpi w^2)^{1/2}, ] since (w(x)+w(y))^2le2(w(x)^2+w(y)^2) and the total outgoing rate is at most R. If sum_Cpi w^2 vanishes the estimate is immediate; otherwise divide and square. The sum of the energies of the positive and negative parts is at most mathfrak D(u).frakD(u). The weighted mean minimizes squared error, proving (eq:14). Step 3: entropy dissipation for the full chain.: For a fixed starting state x, set f_t^x(y)=P_t(x,y)/pi(y). Reversibility together with (a-b)(log a-log b)ge4(sqrt a-sqrt b)^2 for a,b>0 gives the entropy-dissipation estimate cite[Lemmas~2.5,~2.7]{DiaconisSaloffCoste1996} int_0^Tsum_xpi(x),mathfrak D(√(f_t^x))frakD(√(f_t^x)),dt letfrac14 H(pi). Indeed, the derivative of sum_ypi(y)f_t^x(y)log f_t^x(y) is [ -sum_{{y,z}}q(y,z) (f_t^x(y)-f_t^x(z))(log f_t^x(y)-log f_t^x(z)). ] The initial entropy is log(1/pi(x)) and the final entropy is nonnegative. Averaging in xsimpi and integrating proves (eq:15). Vanishing densities and reducible chains cause no problem: start with (1-eta)f_0^x+eta and pass to eta↓0, or calculate on communicating classes. Step 4: averaging fixed-state distances.: Combining (eq:14) and (eq:15) supplies some tin(0,T] with [ sum_xpi(x)r_x^2le(R,H(pi))/(2TB^2), r_x^2=sum_Cpi(y)(√(f_t^x(y))-a_x)^2, a_x=frac1{pi(C)}sum_Cpi(y)√(f_t^x(y)) . ] Cauchy--Schwarz and a_x^2pi(C)lesum_Cpi f_t^xle1 imply [ sum_Cpi|f_t^x-a_x^2|le2r_x . ] Also, total normalization gives |sum_Cpi(f_t^x-1)|lesum_{C^{rm c}}pi(f_t^x+1). Comparing f_t^x first with a_x^2 and then with 1 on C, and adding the complement, yields [ sum_Ωpi|f_t^x-1| le4r_x+2sum_{C^{rm c}}pi(f_t^x+1). ] The last sum before its factor two has average 2pi(C^{rm c}) over xsimpi, because stationarity gives sum_xpi(x)f_t^x(y)=1. Dividing by two for total variation and applying Cauchy--Schwarz in x proves the upper bound at time t. Contraction under P_{T-t} gives the same bound at the prescribed time T, establishing (eq:averaged-mixing). ## Proof of the main theorem Proof. [Proof of Theorem~thm:main] On the disorder event of Proposition~prop:cuts, apply Lemma~lem:pruning to the SK chain with [ ε=n^{-d}, B=exp(-n^{1-b/20}), R=n . ] The last choice is valid because there are exactly n rate-one update clocks, so the total jump rate at any vertex is n. Also H(pi_G)le nlog2, since the state space has 2^n elements. Hence [ sum_xpi_G(x)bigl|P_T^G(x,cdot)-pi_Gbigr|_{rm TV} lefrac{√(2log2);n}{Bsqrt T}+2n^{-d}. ] Take [ T=exp(n^{1-b/40}). ] Then the first term tends to zero, because [ log(n)/(Bsqrt T) =log n+n^{1-b/20}-tfrac12 n^{1-b/40}→-∞ . ] The flow event (eq:11) itself has disorder probability tending to one. Markov's inequality therefore shows that the Gibbs mass of fixed states whose distance at time T exceeds 1/4 tends to zero in disorder probability. The added diagonal changes neither this Gibbs law nor the transition rates, so the same statement holds for the model H^J_n in (eq:H). This proves Theorem~thm:main with [ c_beta=(b)/(40)=frac1{40000000}. ] All threshold dimensions may depend on the fixed beta>1. ## Reading the final exponents For orientation, here is the chain of inequalities that governs the constants, all in the ``1-exponent'' scale: [ 10b ll b/4 ll b/2 ll b/20 ll d =100b ll 2dll fll 1 . ] The two competing probabilities in Section~sec:connections were the coverage probability exp(-2n^{1-2d}) and the sequence failure probability exp(-n^{1-d/2}): the ratio of their exponents is 2d versus d/2, and d/2ll d<2d gives the gap. The mixing time T=exp(n^{1-b/40}) sits below both because b/40ll b/20, so the exponential exp(-n^{1-b/20}) flow bound has enough time to accumulate. 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