Model and main theorem

Let \(n\ge2\), and let \(J_{ij}\), \(1\le i<j\le n\), be independent standard Gaussian variables. For \(\sigma\in\{-1,1\}^n\), define \[\begin{equation*} \tag{H} H_n^J(\sigma)=\frac1{\sqrt n}\sum_{i<j}J_{ij}\sigma_i\sigma_j, \qquad \pi_{n,\beta}^J(\sigma) =\frac{\exp(\beta H_n^J(\sigma))}{Z_{n,\beta}^J}. \end{equation*}\] The external field is zero. Each site has an independent rate-one clock. When the clock at site \(i\) rings, the new spin equals \(s\in\{-1,1\}\) with conditional probability \[\frac{\exp(\beta s h_i(\sigma))}{2\cosh(\beta h_i(\sigma))}, \qquad h_i(\sigma)=\frac1{\sqrt n}\sum_{j\ne i}J_{ij}\sigma_j,\] where \(J_{ji}=J_{ij}\) and \(J_{ii}=0\). Such an update may leave the configuration unchanged. Denote the resulting continuous-time transition law from a fixed \(\sigma\) by \(P_t^{J,\beta}(\sigma,\cdot)\), and set \[\begin{equation*} \tag{T} T_{n,\beta}^J(\sigma) =\inf\left\{t\ge0: \left\|P_t^{J,\beta}(\sigma,\cdot)-\pi_{n,\beta}^J\right\|_{\mathrm{TV}} \le\frac14\right\}. \end{equation*}\]

Equivalently, in probability over \(J\), the Gibbs mass of states satisfying the displayed bound tends to one. The state \(\sigma_0\) is realized before the transition law and its distance from equilibrium are formed. If the initial state were averaged into that transition law, stationarity would instead make the distance zero at every time. The exponent above is not optimized, and the dimension at which the estimates become effective may depend on the fixed \(\beta\). All times in this paper use the continuous-time convention just specified.

Earlier mixing results and proof ingredients

At high temperature, functional inequalities have yielded polynomial mixing bounds from every starting state. Eldan, Koehler and Zeitouni proved such a bound for \(\beta<1/4\) using a spectral condition on the interaction matrix [8]. More recently, Wang proved optimal-order Glauber mixing for every fixed \(\beta<1/2\) [18], and Boban, Li and Oveis Gharan extended polynomial mixing in zero field to \(\beta<1/2+\varepsilon_0\) for an absolute \(\varepsilon_0>0\) [2]. These results concern worst-start mixing in a high-temperature range. In the low-temperature setting, Sellke explicitly distinguished the worst-start obstruction from the possibility of faster mixing after Gibbs-distributed initialization [16]. Theorem 1 addresses this latter question throughout the range of fixed \(\beta>1\).

The equilibrium input that drives the proof is stationarity along an entire interval of scalar overlaps. The variational analysis of the Parisi functional developed by Chen [6] and by Jagannath and Tobasco [10] provides the scalar stationarity conditions on the support of the minimizing measure. Zhou established full interval support immediately above the transition [19]. For the whole range of fixed \(\beta>1\), we use the full interval conclusion of Lopatto [12], specifically version 3. Full support lets us apply the stationarity identity at every overlap between zero and the top support point. Infinite support alone would not supply this input.

Two other equilibrium results enter with different purposes. Guerra’s finite-volume interpolation bound [9] controls the expected log partition function from above. The spatial regularity and step-approximation stability of the scalar Parisi equation [1] justify the diffusion and its calculus. Section 3 states these inputs in the exact normalization used here and derives the additional estimates beyond the support endpoint.

The proposal construction is related to established Gaussian-query methods. Bolthausen’s iterative construction of TAP solutions uses successive Gaussian conditioning and orthogonalization [4]. Montanari’s incremental approximate message passing algorithm uses a Parisi diffusion and normalized magnetization increments to optimize the SK Hamiltonian [14]. Our proposal uses exact finite-dimensional Gaussian conditioning to establish quantitative entropy and likelihood comparisons with the Gibbs joint law.

The comparison in the reverse direction uses transport under a strongly log-concave reference measure [3], followed by Gaussian smoothing [13]. We give the necessary conditional proofs in Section 5. Finally, the conversion from flow to mixing uses the coarea proof of the reversible Cheeger bound [11] and entropy dissipation [7]. Our finite-chain argument retains a set of large stationary mass before making this conversion, so that it only requires information about cuts separating appreciable masses.

How the proof reaches a typical fixed state

We first construct a random spin configuration correlated with the Gaussian interaction matrix. Its coordinates follow a scalar magnetization martingale. The martingale continues beyond the Parisi support until the average coordinate uncertainty is small. During this continuation, the matrix-query noise is mixed with an independent Brownian noise. An exact scalar identity balances the entropy paid through the matrix channel against the energy gained by the proposal. Sections 3 and 4 turn this identity into two estimates: sublinear forward relative entropy, and an upper likelihood bound outside an exponentially unlikely event.

Forward relative entropy by itself does not guarantee that a proposal reaches a prescribed set of positive Gibbs mass. To obtain that property, Section 5 conditions the Gibbs joint law on the spin. The remaining disorder law is strongly log-concave. Smoothing the disorder then reverses the entropy comparison and gives a lower bound on the proposal probability of every set whose Gibbs mass is appreciable, for the realized target disorder and smoothing base.

Section 6 joins two proposals generated independently conditional on that base. Longer, fine-horizon proposals supply the endpoint coverage; shorter, coarse-horizon proposals have the rarer failures needed along the connecting sequence. Near-pinning of the magnetizations joins these two scales. Rotating the Gaussian inputs of the coarse proposals produces small consecutive Hamming distances. If a cut separating appreciable Gibbs masses had very small stationary flow, its enlarged vertex boundary would have tiny Gibbs mass. The likelihood bound makes the intermediate proposals avoid that boundary, while coverage gives a positive chance that the two endpoints lie on opposite sides. These conclusions contradict the elementary fact that a short-step sequence crossing a cut must approach its boundary; Figure 1 depicts this last step.

It remains to turn the resulting flow estimate into a mixing statement. Section 7 proves a finite-chain lemma that removes less than a prescribed small stationary mass and obtains a Poincaré inequality on the remainder. Entropy dissipation and stationarity then bound the average of the fixed-state total-variation distances. The argument does not require a trajectory to stay in the retained set. This is the final reason that rare slow starting states do not prevent the conclusion of Theorem 1.

Exact equilibrium inputs and normalization

Let \(\mu_\beta\) be a minimizing Parisi probability measure for zero-field Gaussian SK at the fixed finite \(\beta>1\). We use the following external results: Guerra’s finite-volume pressure upper bound [9]; spatial regularity and stability under step approximation for the Parisi equation [1]; and full interval support with scalar stationarity [12]. In particular, \[\operatorname{supp}(\mu_\beta)=[0,q_*],\qquad 0<q_*<1.\] The cited support theorem is used only for this zero-field Gaussian pair-interaction model and for each fixed \(\beta>1\).

Use scalar variance time \(v=\beta^2s\), and put \(\alpha_v=\mu_\beta([0,v/\beta^2])\) for \(0\le v\le\beta^2\). Let \(\Phi\) solve the backward Parisi equation \[\Phi_v=-\tfrac12(\Phi_{yy}+\alpha_v\Phi_y^2), \qquad \Phi(\beta^2,y)=\log(2\cosh y).\] Derivatives of \(\Phi\) are indicated by subscripts. Equivalently, if \(u\) is the scalar solution in [12], then \(\Phi(v,y)=u(v/\beta^2,y)+\log2\). Combining the source’s independent ordered pair couplings gives the unordered pair coefficient \(\beta/\sqrt n\) used here, plus a centered diagonal term. Thus its scalar normalization agrees with the completed Hamiltonian.

Set \(v_*=\beta^2q_*\). In these units the pressure functional and the finite-volume upper bound read \[\mathcal P=\Phi(0,0)-\frac1{2\beta^2} \int_0^{\beta^2}v\alpha_v\,dv, \qquad \mathbb E F(G)\le n\mathcal P.\] The centered diagonal does not affect this expected pressure. Since \(\alpha_v=1\) above \(v_*\), the Gaussian recursion gives \[\Phi(v_*,y)=\log(2\cosh y)+\frac{\beta^2-v_*}{2}.\] Only the pressure upper bound is needed below.

Continue the endpoint formula for the potential to all \(v>v_*\) by \[\Phi(v,y)=\log(2\cosh y)+\frac{\beta^2-v}{2},\qquad \alpha_v=1.\] This agrees with the original solution up to \(\beta^2\) and satisfies the same PDE for arbitrarily large \(v\). For every \(v\ge0\), put \(m(v,y)=\Phi_y(v,y)\) and \(a(v,y)=\Phi_{yy}(v,y)\); thus \(m(v,y)=\tanh y\) and \(a(v,y)=1-\tanh^2y\) when \(v\ge v_*\). Let \(Z\) be standard Brownian motion and solve \[dY_v=\alpha_vm(v,Y_v)\,dv+dZ_v,\qquad Y_0=0.\] The drift is bounded and spatially Lipschitz. We abbreviate its evaluated scalar functions by \(m_v=m(v,Y_v)\) and \(a_v=a(v,Y_v)\). The cited stationarity identity, the differentiated PDE, and Itô’s formula give \[m_0=0,\qquad dm_v=a_v\,dZ_v,\qquad \mathbb E m_v^2=v/\beta^2\quad(0\le v\le v_*).\] Full interval support is what makes the last identity available for every time in \([0,v_*]\); the variational identity on the support is also given in [10]. The martingale equation continues beyond \(v_*\) by direct Itô calculus. Spatial derivatives are continuous under step approximation, so none of these assertions requires time differentiability at a CDF atom.

We will repeatedly use \[0<a(v,y)\le1-m(v,y)^2,\qquad |m(v,y)|<1.\] Here is the derivative bound in a form suited to step approximation. One step with CDF height \(\lambda\in[0,1]\) and variance \(\delta>0\) takes a profile \(f\) to \[(Tf)(y)=\lambda^{-1}\log\mathbb E e^{\lambda f(y+\sqrt\delta U)},\qquad U\sim N(0,1),\] with \(Tf(y)=\mathbb E f(y+\sqrt\delta U)\) when \(\lambda=0\). Write \(\mathbb E_\lambda\) for the Gaussian expectation tilted by \(e^{\lambda f(y+\sqrt\delta U)}\) and normalized to have total mass one. Differentiating gives \[(Tf)'=\mathbb E_\lambda f',\qquad (Tf)''=\mathbb E_\lambda f''+ \lambda\operatorname{Var}_\lambda(f').\] If \(0<f''\le1-(f')^2\), these formulas imply \[0<(Tf)''\le1-\bigl((Tf)'\bigr)^2 -(1-\lambda)\operatorname{Var}_\lambda(f') \le1-\bigl((Tf)'\bigr)^2.\] Inducting from \(f(y)=\log(2\cosh y)\) proves the bound for step CDFs; scalar stability passes the non-strict bound to the minimizing CDF. The positive lower bound for \(\Phi_{yy}\) in [12] supplies strict positivity in the limit. Hence \(m\) is strictly increasing and bounded by one in absolute value, so \(|m|<1\). On the extension these assertions follow from \(\tanh\).

Tail estimates for the extended diffusion

Define the deterministic functions \[p(v)=\mathbb E a_v,\qquad r(v)=\mathbb E a_v^2, \qquad \rho(v)=\beta\sqrt{r(v)}.\] The martingale second-moment identity and continuity give \(r(v)=1/\beta^2\) on \([0,v_*]\), including the endpoints. The next lemma controls the continuation and shows that \(\rho\) is an admissible noise-mixing coefficient.

Balancing scalar entropy and energy

Let \(\mathcal H(z)\) be the entropy of a sign with mean \(z\): \[\mathcal H(z)=-\frac{1+z}{2}\log\frac{1+z}{2} -\frac{1-z}{2}\log\frac{1-z}{2},\] with the usual continuous values at \(z=\pm1\). Define, for \(V>v_*\), \[e(V)=\int_0^V\sqrt{r(v)}\,p(v)\rho(v)\,dv, \qquad j(V)=\frac12\int_0^V\rho(v)^2 \mathbb E\frac{a_v^2}{1-m_v^2}\,dv.\] The quantity \(e(V)\) will be the energy per spin of the matrix proposal; \(j(V)\) will measure the information acquired through its matrix channel. The independent noise introduced later accounts for the factor \(\rho(v)^2\) in this information cost.

Up to \(v_*\), Itô’s formula, the PDE and stationarity give \[\begin{aligned} \mathbb E[Y_{v_*}m_{v_*}] &=\int_0^{v_*}\bigl(p(v)+\alpha_v v/\beta^2\bigr)\,dv,\\ \mathbb E\Phi(v_*,Y_{v_*}) &=\Phi(0,0)+\frac12\int_0^{v_*}\alpha_v v/\beta^2\,dv. \end{aligned}\] These formulas also follow by step approximation with bounded first and second spatial derivatives. Substitute the endpoint value of \(\Phi\) and use \(\mathcal H(\tanh y)=\log(2\cosh y)-y\tanh y\) to obtain \[\int_0^{v_*}p(v)\,dv+\mathbb E\mathcal H(m_{v_*}) =\mathcal P-\frac{\beta^2}{4}(1-q_*)^2. \tag{2}\]

The terminal error in [eq:3] is at most \(C_\beta e^{-V}\). We will take \(V\) proportional to \(\log n\), so that this error is a negative power of the dimension while the scalar spin is nearly pinned.

Construction and conditional Gaussian laws

Successive Gaussian revelations along orthogonalized directions occur in the iterative TAP construction [4]. Incremental approximate message passing uses related Parisi-guided normalized increments with asymptotic orthogonality [14]. Here we give an exact conditional-law argument for our finite mesh, including the auxiliary noises that complete each increment.

Fix \(0<l\le1/100\) and define \[h=n^{-1/16},\qquad K=\lceil l\log n/h\rceil,\qquad t_k=kh,\qquad V=t_K.\] All assertions below are for sufficiently large \(n\), so \(K<n\) and \(V>v_*\). The disorder \(G\) has its ordinary Gaussian law. We construct a vector process with coordinates following the scalar SDE, using \[dZ=\rho(v)\,dB+\sqrt{1-\rho(v)^2}\,dD. \tag{4}\] Here \(D\) is an independent auxiliary Brownian motion. The stream \(B\) is constructed from matrix queries and auxiliary completions so that, after averaging over the disorder, \((B,D)\) has independent standard Brownian coordinates. The weight \(\rho\) equals one up to \(v_*\) and is nonincreasing afterward. The continuation therefore reduces the share of the query-generated stream while keeping unit variance in \(Z\). Put \(M(v)_i=m(v,Y_{v,i})\).

On \([0,h]\), generate \(B\) entirely from independent auxiliary Brownian noise. For each subsequent interval \([t_k,t_{k+1}]\), \(1\le k<K\), the previous scalar increment determines \[s_k^2=\int_{t_{k-1}}^{t_k}r(v)\,dv, \qquad T_k=\frac{M(t_k)-M(t_{k-1})}{s_k}.\] Project \(T_k\) off the preceding query directions \(u_1,\ldots,u_{k-1}\), and normalize the residual to obtain \(|u_k|_n=1\). If the residual is zero, choose a new direction measurably from the past: project the coordinate basis vectors in their fixed order, take the first nonzero projection, and normalize it. Since \(k<n\), such a direction exists. Let \(\Pi_{\le k}\) be the orthogonal projection onto the span through \(u_k\), and write \(\Pi_{>k}=I-\Pi_{\le k}\).

Query the column \(Wu_k\). For an independent standard Gaussian vector \(z_k\), set the endpoint of the next Brownian increment to be \[g_k=h^{-1/2}\bigl(B(t_{k+1})-B(t_k)\bigr) =\Pi_{>k}Wu_k+\Pi_{\le k}z_k.\] Fill the interval with linear interpolation between its endpoints plus an independent centered Brownian bridge. Every new bridge and completion noise is fresh. Deterministic-coefficient integrals against this filled path may equivalently be obtained by conditioning Brownian motion on its endpoint. At time \(V\), draw the spins independently given the path, with means \(M_i(V)\). Call the marginal law of \((G,x)\) obtained this way \(Q_l\).

To verify the stated Brownian law, condition on the previously revealed columns and the auxiliary inputs already used. Gaussian orthogonal invariance leaves the unqueried block GOE on the orthogonal complement, with the original variance scale. The next direction 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. Therefore \(\Pi_{>k}Wu_k\) is a fresh Gaussian with covariance \(\Pi_{>k}\), and \[d_k=\langle u_k,Wu_k\rangle_n\sim N(0,2/n)\] conditionally before the column is revealed. The independent completion has covariance \(\Pi_{\le k}\), so \(g_k\) is a fresh standard Gaussian vector. Induction, followed by the independent bridge filling, proves the assertion about \((B,D)\) in [eq:4].

In particular, after forgetting the matrix, the coordinate paths and their final spins are independent and identically distributed. The conditional mean of \(x\) given the Brownian streams through time \(v\) is \(M(v)\). 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. These facts concern marginals; the proposal spin remains correlated with its matrix.

Entropy and likelihood estimates

The following 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.

Why smoothing reverses the entropy comparison

We prove \[{\rm KL}(\mathcal SP\Vert\mathcal SQ_f) \le\delta^{-1}{\rm KL}(Q_f\Vert P). \tag{8}\] The two unsmoothed laws have the same uniform spin marginal. Given \(x\), the conditional disorder density under \(P\) has negative logarithm, up to a normalizing constant, \[U_x(G)=\tfrac12\|G\|^2-\beta\widehat H_G(x)+F(G).\] Its Hessian is at least the identity: the Hamiltonian is linear in \(G\), while \(F\) is a convex log-sum-exp of linear functions.

We need the following consequence of this convexity. A law \(\mu\) with density proportional to \(e^{-U}\), where \(\nabla^2U\ge I\), and a law \(\lambda\ll\mu\) admit couplings whose mean squared displacement is at most \(2{\rm KL}(\lambda\Vert\mu)\), or tends to that bound. This is the quadratic transport inequality; its triangular-transport form appears in [3]. We recall the proof for the finite-dimensional densities used here.

First restrict to a box and suppose both densities are smooth and strictly positive up to its boundary. Map \(\mu\) to \(\lambda\) by successive increasing conditional quantile maps. The resulting triangular map \(T\) has positive diagonal derivatives, whose product is its Jacobian determinant. The entropy formula is \[{\rm KL}(\lambda\Vert\mu) =\int\bigl(U(T(z))-U(z)-\log\det\nabla T(z)\bigr)\,d\mu(z).\] Strong convexity bounds the first difference below by \(\nabla U(z)\cdot(T(z)-z)+\tfrac12\|T(z)-z\|^2\). Integration by parts replaces the linear term by \(\operatorname{div}(T-\mathrm{id})\). There is no boundary term: on a face where the \(i\)th coordinate is an endpoint of the box, its conditional quantile map fixes that endpoint, so the normal component \(T_i-z_i\) vanishes. Tangential motion does not matter. The remaining Jacobian terms are nonnegative because \(z-1-\log z\ge0\) for every positive diagonal derivative \(z\). This proves the coupling bound on the box.

The densities needed here satisfy the approximation requirements. For fixed \(n\), the conditional proposal density is bounded by a constant times Gaussian density, since its spin probability is at most one. Truncate both conditional laws to increasing boxes and normalize them. On each box approximate \(\lambda\) by smoothing and mixing with a small positive constant density. Its density is bounded, and the smooth reference density is bounded away from zero on the box, so their relative entropies with respect to the reference converge. The original laws have Gaussian tails up to constants for fixed \(n\), and finite relative entropy. Thus the truncated entropies converge; tightness and lower semicontinuity pass the coupling bound to the original laws.

Apply this result with \(\mu=P(\cdot\mid x)\) and \(\lambda=Q_f(\cdot\mid x)\). For any coupling of two disorder values \(g,\widetilde g\), the Gaussian smoothing kernels satisfy \[{\rm KL}\bigl(N(\sqrt{1-\delta}\,g,\delta I) \Vert N(\sqrt{1-\delta}\,\widetilde g,\delta I)\bigr) =\frac{1-\delta}{2\delta}\|g-\widetilde g\|^2.\] The convolution comparison is also recorded in [13]. Joint convexity of entropy bounds the divergence between the mixtures by this averaged kernel quantity. Average over the common spin marginal and use the entropy chain rule. The transport bound then proves [eq:8].

Coverage conditional on a target and its base

The smoothed entropy comparison is still a joint-law statement. We now convert it to a statement for the original Gibbs law at a realized target disorder. The following proposition records both the entropy estimate and the coverage consequence used in the next section.