Dynamics of the spherical Sherrington–Kirkpatrick model, Part I: Foundations and tools

Part I: Foundations and tools

AUTHORS’ NOTE

With the exception of this note and small edits to the abstract, the entire document is machine-written. We have taken care to supervise this writing to ensure that the proofs meet a minimum standard of readability and that previous literature is appropriately cited. However, the resulting document remains lacking in exposition and overall coherence. We are nonetheless releasing this manuscript on Hexagon to disseminate the result as quickly as possible, because we believe it may be of interest to the mathematics and physics communities. We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.

For publication, the manuscript is divided into two volumes: Part I: Foundations and tools (Sections 1–12) and Part II: Mixing profiles and critical dynamics (Sections 13–21). Section, theorem, equation, and citation numbers are shared across the volumes, and each volume includes the complete bibliography.

P.L. was partially supported by NSF grant DMS-2450004.

Introduction

The spherical Sherrington–Kirkpatrick model is the simplest mean-field spin glass. Its configurations are points $x$ of the sphere $S_N = \{x \in\mathbb{R}^N : |x|^2 = N\}$, and its energy is a quadratic form in $x$ whose coefficients form a random symmetric matrix, so that its equilibrium can be analyzed exactly through the spectrum of that matrix. The Langevin diffusion on $S_N$ that is reversible with respect to the Gibbs measure has been studied in physics for more than forty years, mostly through the evolution of observables such as the energy and two-time correlation functions in the limit of infinite system size.

The goal of this paper is to describe how the law of this diffusion approaches equilibrium when the dimension $N$ is large and the disorder is fixed. We start the dynamics from the uniform distribution on $S_N$, independently of the disorder, and measure its distance to the Gibbs measure in total variation. We determine the asymptotic behavior of this distance as a function of time at fixed temperatures above and below the critical one, in the crossover to infinite temperature, along every mesoscopic approach to the critical temperature from either side, and in the critical window. In the regimes treated here outside the critical window we find cutoff, except in the part of the crossover to infinite temperature where $N\beta^2$ stays bounded; there the distance relaxes over bounded times. In each case we identify the center and the window of the transition and its limiting shape, which we call the mixing profile. In the critical window the mixing profile is random and determined by the edge of the spectrum of the disorder, and there is no cutoff. We also show that the critical mixing profiles match those of the two mesoscopic regimes far from the critical temperature.

The model and its Langevin dynamics

Spherical models. Berlin and Kac introduced the spherical model of a ferromagnet in [28], in which the Ising spins of a lattice model are replaced by real variables subject to a single constraint on the sum of their squares. Lewis and Wannier considered the mean spherical model, in which this constraint holds only in expectation [107, (1)–(2), p. 683], and Stanley identified the spherical model with the limit of classical vector spin models as the number of spin components tends to infinity [140]. Sherrington and Kirkpatrick introduced the mean-field Ising spin glass with independent Gaussian couplings [132]. Kosterlitz, Thouless, and Jones introduced its spherical version, whose couplings form a random symmetric matrix, and solved its thermodynamics through the spectrum of that matrix [97].

We use the following normalization. Let $W_N$ be an $N \times N$ matrix from the Gaussian orthogonal ensemble (GOE), that is, a real symmetric matrix whose entries on and above the diagonal are independent with $W_{ij} \sim\mathcal{N}(0,(1+\mathbf{1}_{\{i=j\}})/N)$ for $i \leq j$. Let $\lambda_1 \geq\lambda_2 \geq\cdots\geq\lambda_N$ be its eigenvalues, and let $\nu_N$ be the uniform probability measure on $S_N$. For an inverse temperature $\beta\geq0$ we define the Hamiltonian and the Gibbs measure by

where $Z_{N,c}$ is the normalizing constant and $c=\sqrt{2}\,\beta$. Throughout the paper we parameterize the temperature by the rescaled inverse temperature $c$, so that increasing $c$ lowers the temperature. The critical value is $c=1$, that is, $\beta_c=1/\sqrt{2}$. Since its density with respect to $\nu_N$ is the exponential of a quadratic form, $\pi_{N,c}$ is a Bingham distribution [31]. Because $-W_N$ has the same law as $W_N$, negative inverse temperatures lead to the same results with the roles of the top and the bottom of the spectrum exchanged, and it suffices to consider $\beta\geq0$.

We work in an orthonormal eigenbasis of $W_N$ and write $x_1,\ldots,x_N$ for the coordinates in this basis, the eigencoordinates. The Gibbs measure has the following Gaussian representation. Let $z>c\lambda_1$ be the unique solution of

and set $p_i=z-c\lambda_i$ and $q_i=1/p_i$. Since $-\sum_i p_i x_i^2/2=-zN/2+(c/2)\sum_i\lambda_i x_i^2$ on $S_N$, the density of a centered Gaussian vector with independent coordinates of variances $q_1,\ldots,q_N$ is proportional to the Gibbs density on $S_N$. The gradient of $|x|^2$ has constant length on $S_N$, and it follows that $\pi_{N,c}$ is the law of this Gaussian vector conditioned on $|x|^2=N$, where the choice of $z$ in (2) makes the expected squared norm of the unconditioned vector equal to $N$. The numbers $z$, $p_i$, and $q_i$ are the Lagrange multiplier, the precisions, and the variances of Definition 3.3. Equation (2) is the saddle-point equation in the solution of Kosterlitz, Thouless, and Jones [97]; see [6, Lemma 1.3] for a modern account of their contour-integral formula for $Z_{N,c}$. The representation of Bingham distributions by conditioned Gaussian vectors is classical in directional statistics [98], and for spherical models it appears in [5, Proposition 2.7] and, in the critical regime of the present model, in [62, Section 1.2].

As $N \to\infty$, the empirical spectral measure of $W_N$ converges to the semicircle law $\rho_{\mathrm{sc}}(\lambda)\,d\lambda=(2\pi)^{-1}\sqrt{4-\lambda^2}\,d\lambda$ on $[-2,2]$ [152, 153]; for convergence in probability in the present normalization, see [4, Theorem 2.1.1, p. 7]. Also, $\lambda_1$ converges to $2$ by the edge estimates recalled in Lemma 4.2. For $c < 1$ we have $z \to1 + c^{2}$ and the smallest precision $d = p_{1} = z - c\lambda_{1}$ converges to $(1 - c)^{2}$, so that all variances $q_i$ remain bounded. For $c > 1$ we have $d \to0$, and the first eigencoordinate carries a macroscopic fraction of the squared norm under $\pi_{N,c}$, in the sense that $x_{1}^{2}/N \to1 - 1/c$ in probability. This is the condensation of the spherical model onto the top eigenvector [97], [51, Section III]. At low temperature the Gibbs measure is therefore concentrated on two antipodal regions in which $x_1$ is close to $\sqrt{(1 - 1/c)N}$ or to $-\sqrt{(1 - 1/c)N}$. The free energy has a phase transition at $c = 1$ [97]; we refer to [47] for the replica analysis of general spherical $p$-spin models and to [143] for a proof of the free-energy formula for spherical models with even interactions, which include the present one.

Langevin dynamics. The Langevin dynamics of the model is the diffusion on $S_N$ with generator

where $\Delta_{S_N}$ and $\nabla_{S_N}$ are the Laplace–Beltrami operator and the gradient on $S_N$. Its noise coefficient is $\sqrt{2}$, and it is reversible with respect to $\pi_{N,c}$. We write $(P_t)_{t\geq0}$ for its semigroup, so that $\nu_NP_t$ is the law at time $t$ of the diffusion started from the uniform distribution. In the eigenbasis the drift is linear apart from the term that keeps the process on the sphere. However, the noise is projected onto the tangent space of $S_N$, and for finite $N$ the eigencoordinates are coupled and are not independent Ornstein–Uhlenbeck processes.

This dynamics has a long history in physics. Shukla and Singh studied relaxation in the spherical spin-glass model [133]. Ciuchi and de Pasquale studied nonlinear relaxation and ergodicity breaking in random anisotropy spin glasses with many spin components [44]. Cugliandolo and Dean solved the dynamics completely in the thermodynamic limit, for quenches from high temperature and for more general temperature protocols [51]. They diagonalized the coupling matrix and reduced the spherical constraint to a scalar Volterra equation for a time-dependent Lagrange multiplier [51, (II.2)–(II.7)], and from its solution they computed correlation and response functions and showed that the dynamics ages at low temperature. Zippold, Kühn, and Horner refined the time scales that separate stationary relaxation from aging and compared the model with spherical ferromagnets [156, Sections III–IV]. Cugliandolo and Kurchan developed a microscopic aging analysis for spherical $p > 2$ spin glasses after taking the thermodynamic limit [50, Introduction], and the quadratic model behaves like a coarsening system [52, Section IV.C]. Caltagirone, Parisi, and Rizzo computed the off-equilibrium correlations at the critical temperature [38, Section III]. On the mathematical side, Ben Arous, Dembo, and Guionnet proved aging for the Langevin dynamics of the spherical Sherrington–Kirkpatrick model with a soft spherical constraint [26]. The same authors derived rigorously the Cugliandolo–Kurchan equations for the correlation and response functions of spherical spin glasses with a soft spherical constraint [23, Theorem 1.2]. Dembo, Guionnet, and Mazza obtained the limit of these equations under the hard spherical constraint [55, Proposition 1.1], and proved that at sufficiently high temperature the solutions converge to a stationary pair that satisfies the fluctuation-dissipation relation and decays exponentially [55, Theorem 1.3]. These analyses take the thermodynamic limit first and then study the resulting correlation and response functions, including their long-time behavior.

Finite-size effects enter through the edge of the spectrum. Rodgers and Moore identified the first-gap barrier between the two antipodal low-temperature regions [124, Section 2, (25)–(26)]; see also [73, Section 4]. Fyodorov, Perret, and Schehr showed that at zero temperature the relaxation crosses over at times of order $N^{2/3}$ from the algebraic decay of the thermodynamic limit to an exponential decay governed by the random gap $\lambda_1-\lambda_2$, in a regime where observables are no longer self-averaging [73, Section 3.1]. Barbier, de Freitas Pimenta, Cugliandolo, and Stariolo studied these finite-size effects further, including the escape from initial conditions concentrated near saddle points [13]. For gradient descent on the sphere started from the uniform distribution, which is the dynamics at zero temperature, Yu showed that the time needed to reach a fixed overlap with an extreme eigenvector lies between orders $N^{2/3}$ and $N^{2/3}\log N$ with probability at least $1-\delta$, for each fixed $\delta> 0$ [155, Section 3, Theorem 3.3]. That result concerns a hitting time, while our results concern the law of the diffusion at positive temperature in total variation. In an approximation that neglects finite-size fluctuations of the spectral density, Cugliandolo, Dean, and Yoshino found a relaxation time of order $N(\beta-\beta_c)$ for the linear susceptibility at low temperature [53, Sections 1.1 and 4.3].

The question. In this paper we study the law of the diffusion itself at finite $N$. We start the diffusion from $\nu_N$, independently of $W_N$, and measure its distance to equilibrium by

and the supremum is over Borel sets $A\subset S_N$. We ask how $D_N(t)$ behaves as a function of $t$ when $N$ is large, and how the answer depends on the temperature.

We comment on three features of this formulation. First, total variation controls the law of the entire configuration, so that once $D_N(t)$ is small the law of every measurable observable is close to its equilibrium law, including observables that depend on all $N$ coordinates; it is also the standard distance in the theory of mixing times [105]. Second, the uniform distribution is the Gibbs measure at $\beta=0$, and the dynamics started from $\nu_N$ describes a quench from infinite temperature. In the eigenbasis a uniform point has coordinates $\sqrt{N}g_i/\lVert g\rVert$, where $g_1,\ldots,g_N$ are independent standard Gaussian variables. Several of the computations cited above instead model the quench by giving every eigenmode the same deterministic initial amplitude [51, (III.1), 73, (6), 13, (12)]. For a fixed matrix this is a different initial condition, and § 2.6 explains why the difference matters at low temperature. Third, the law $\nu_NP_t$ is averaged over the initial point and the Brownian motion but not over the disorder, so that $D_N(t)$ is a random variable that depends on $W_N$ only, and our results are statements in probability or in distribution with respect to $W_N$.

Cutoff and limit profiles