Exact asymptotics for spherical Sherrington-Kirkpatrick energy dynamics and extreme-eigenvector hitting times beyond the quadratic constraint, i.i.d. initialization, and logarithmic slack

The spherical Sherrington-Kirkpatrick (SK) spin glass places $N$ continuous spins on a sphere and couples them through a Wigner random matrix; its rotational symmetry makes the Langevin dynamics of this benchmark glassy system explicitly solvable in the thermodynamic limit. The source paper, Yu (arXiv:2401.03668, 2024), proved that, for i.i.d. initial laws and any admissible soft-constraint potential $f$, the energy $H_N(t)=N^{-1}X_t^{\top}JX_t$ converges almost surely to a deterministic limit $H(t)$ solving an integro-differential equation; for the quadratic soft spherical constraint $f(x)=cx^2/2$ with $K(0,0)=1$ he computed $\lim_{t\to\infty}H(t)$ explicitly, including a dynamical phase transition at the critical inverse temperature $\beta_c=c/4$ with a jump discontinuity of size $2/c$. The same work proved that the hitting times of spherical gradient descent and of power iteration to a fixed overlap $\epsilon$ with an extreme eigenvector of a Wigner matrix lie between order $N^{2/3}$ and $N^{2/3}\log N$ with high probability under a finite fourth-moment condition. The surviving open problem has three linked components: (1) the long-time energy limit and phase structure for general constraint potentials $f$; (2) the same explicit energy asymptotics for non-i.i.d., disorder-dependent, Gibbs, and signal-bearing initial data; (3) sharp hitting-time asymptotics — limiting distributions, optimal constants, necessity of the $\log N$ factor, and behavior under heavy-tailed entries. No resolution of any of these components is known in the literature to date.

Background

A spin glass is a disordered magnetic system in which random couplings frustrate simple ordering. Its mean-field idealization, the Sherrington-Kirkpatrick (SK) model, assigns to a configuration $X=(X_1,\dots,X_N)$ the random Hamiltonian $H_J(X)=X^{\top}JX$, where $J$ is a normalized Wigner matrix: a symmetric random matrix whose independent upper-triangular entries have mean zero and variance $1/N$ (the Gaussian case is the GOE, whose empirical eigenvalue distribution converges to the semicircle law on $[-2,2]$). In the spherical SK (SSK) model the spins live on the sphere $\mathbb{S}^{N-1}(\sqrt{N})=\{X\in\mathbb{R}^N:\|X\|^2=N\}$, which makes the model rotation-invariant in law and exactly solvable; it is the standard laboratory for rigorous results on out-of-equilibrium spin-glass dynamics.

To keep the dynamics on the sphere while remaining tractable, the hard constraint is replaced by a soft one enforced through a potential $f$ with $f'$ non-negative and Lipschitz, and one studies the Langevin dynamics at inverse temperature $\beta$: $dX_t^i=\sum_{j=1}^N J_{ij}X_t^j\,dt-f'\Big(\frac{1}{N}\sum_{j=1}^N(X_t^j)^2\Big)X_t^i\,dt+\beta^{-1/2}dW_t^i.$ In the eigenbasis of $J$ — and for quadratic $f$, where the drift is linear — this becomes $N$ coupled Ornstein-Uhlenbeck processes (Gaussian processes with linear restoring drift and constant diffusion) with random drifts given by the eigenvalues of $J$. Ben Arous, Dembo and Guionnet proved that the empirical correlation function $K_N(t,s)=N^{-1}\sum_iX_t^iX_s^i$ converges as $N\to\infty$ to the unique solution of a nonlinear integro-differential equation — the rigorous form of the Cugliandolo-Kurchan equations of glass physics, also called the CHSCK equations in this literature — and analyzed its long-time behavior, giving the first mathematical proof of aging, the phenomenon that the older the system gets, the longer it takes to decorrelate, including a dynamical phase transition. Their analysis covers several initial laws for the correlation function, with later extensions to mixed $p$-spin models and to disorder-dependent and Gibbs-distributed initial data.

Yu (arXiv:2401.03668) extended this program from the correlation function to the energy $H_N(t)=N^{-1}X_t^{\top}JX_t$: for i.i.d. initial laws, $H_N$ converges almost surely to a deterministic $H(t)$ solving an explicit integro-differential equation built from the limiting correlation function, for any admissible $f$. For the quadratic constraint $f(x)=cx^2/2$ with $K(0,0)=1$, the long-time limit is explicit: $\lim_{t\to\infty}H(t)=0$ for $\beta\le\beta_c$ and $\lim_{t\to\infty}H(t)=(4\beta-c)/(2^{5/2}\sqrt{\pi}\,c\beta)+\frac{1}{2}\beta^{-1}$ for $\beta>\beta_c$, where the critical inverse temperature is $\beta_c=c/4$ — a dynamical phase transition with a jump discontinuity of size $2/c$.

The same paper connects the dynamics to algorithms. Spherical gradient descent $dX_t=-\nabla_{\mathbb{S}^{N-1}}H_J(X_t)\,dt$ (the zero-temperature, $\beta=\infty$ dynamics) is a continuous analogue of power iteration $q_k=Jq_{k-1}/\|Jq_{k-1}\|$, the classical iterative method for the dominant eigenvector. For both, started from a uniform random initialization, the hitting time $T_\epsilon$ of overlap $\epsilon$ with the extreme (top or bottom) eigenvector of a normalized Wigner matrix satisfies $C_1N^{2/3}<T_\epsilon<C_2N^{2/3}\log(C_3N)$ with probability tending to one, assuming the entries have a finite fourth moment. Here the overlap $v\cdot X_t$ measures the alignment of the iterate with a target eigenvector $v$, and the $N^{2/3}$ scale comes from random-matrix edge universality: the extreme eigenvalues of a Wigner matrix fluctuate on the $N^{-2/3}$ scale with Tracy-Widom limit law (the universal limiting distribution of the rescaled largest eigenvalue), the gap between the two extreme eigenvalues is of order $N^{-2/3}$, and the finite fourth moment is the standard sufficient condition for this edge behavior (Lee-Yin). For context, the average-case complexity of eigenvector computation has a longer lineage: Kostlan's statistical analysis of dominant-eigenvector calculation, and Deift and Trogdon's universality theorem for the halting times of power and QR eigenvalue algorithms on random sample covariance matrices. Before Yu's work, the indefinite Wigner case with eigenvector-overlap targets had no two-sided complexity bounds; its sharp asymptotics remain unknown, including whether the scaled hitting time has a limiting law and how such a law would combine the Tracy-Widom edge fluctuations, the $N^{-2/3}$ extreme-eigengap, and the $N^{-1/2}$-scale random initial overlap.

Problem Statement

The problem is to remove the three restrictive hypotheses under which the exact asymptotics of the spherical SK dynamics and its algorithmic counterpart are currently proven — the quadratic constraint potential, the i.i.d. initial law, and the logarithmic slack in the hitting-time bounds. It has three components, one for each hypothesis.

Setting. The spherical SK model with Hamiltonian $H_J(X)=X^{\top}JX$ on $\mathbb{S}^{N-1}(\sqrt{N})$, where $J$ is a normalized Wigner matrix; the soft-constraint Langevin dynamics $dX_t^i=\sum_{j=1}^N J_{ij}X_t^j\,dt-f'\Big(\frac{1}{N}\sum_{j=1}^N(X_t^j)^2\Big)X_t^i\,dt+\beta^{-1/2}dW_t^i$ with $f'$ non-negative and Lipschitz; the energy $H_N(t)=N^{-1}X_t^{\top}JX_t$, whose deterministic limit $H(t)$ for i.i.d. initial laws satisfies the integro-differential limit equation proven in the source for any admissible $f$; and, for the algorithmic part, spherical gradient descent on the unit sphere and power iteration $q_k=Jq_{k-1}/\|Jq_{k-1}\|$ from a uniform random start on a normalized Wigner matrix with entries of finite fourth moment, with hitting time $T_\epsilon=\inf\{t\ge 0:|v\cdot X_t|\ge\epsilon\}$ to a fixed overlap $\epsilon\in(0,1)$ with the extreme eigenvector $v$.

Component 1 (general constraint potential). Determine the long-time behavior of $H(t)$ beyond the quadratic $f(x)=cx^2/2$: does $\lim_{t\to\infty}H(t)$ exist for general admissible $f$; what is its value (in closed form, or as an explicit functional of $f$ and $\beta$); and what is the phase structure in $\beta$ — in particular, does a dynamical phase transition persist, what is the critical $\beta_c(f)$, is the transition discontinuous, and how do the threshold and limiting value depend on $f$?

Component 2 (non-i.i.d. initial data). Extend the explicit long-time energy limits and phase boundary beyond i.i.d. initial laws to the initial-data classes known from the correlation-function theory to change the aging behavior. These classes are representative rather than jointly required, and extending the limits to any one of them resolves this component. The classes are rotated (eigenbasis-aligned) initial conditions, disorder-dependent initial conditions such as the top-eigenvector initial condition, Gibbs (equilibrium or quenched) initial laws at temperatures different from the bath, and signal-bearing initial conditions — initial data with non-trivial overlap with extreme eigenvectors, or involving a spiked interaction matrix, i.e. a Wigner matrix plus a low-rank deterministic perturbation whose outlier eigenvalues detach from the semicircle edge.

Component 3 (sharp algorithmic asymptotics). Determine the sharp asymptotics of $T_\epsilon$ (in continuous time for gradient descent, in iteration count for power iteration): the limiting distribution of the appropriately centered and scaled $T_\epsilon$ — in particular whether $N^{-2/3}$-scaled $T_\epsilon$ (with any log-scale centering) has a weak limit, and, if so, what that limit law is; the optimal leading constants in the two-sided bounds; whether the $\log N$ factor in the upper bound is necessary or removable; and the analogue of the $N^{2/3}$ complexity scale when the fourth-moment and edge-universality condition fails (heavy-tailed entries).

Each of the three components is an independent sub-problem that may be resolved on its own. The three components share one objective: replacing the restricted formulas and coarse high-probability bounds of the source by exact characterizations, in each of the directions that the source itself identifies as limited by its hypotheses — components 1 and 2 remove the two hypotheses of the same theorem (the energy phase diagram), and component 3 sharpens the companion algorithmic bounds derived from the same rotational-invariance reduction. The source additionally lists finite-$N$ quantitative error control and a quantitative finite-size link between aging and algorithmic performance as aspirational directions without a precise target; they are out of scope here.

References

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