Parisi's solution of the Sherrington--Kirkpatrick model from synchronous Monte Carlo dynamics

Yoshiyuki Kabashima

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arXiv · 2610.05557

Parisi's solution of the Sherrington--Kirkpatrick model from synchronous Monte Carlo dynamics

Yoshiyuki Kabashima

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Abstract

Sompolinsky and Zippelius obtained Parisi's solution of the Sherrington-Kirkpatrick (SK) model from the relaxational dynamics of soft spins in continuous time, by assuming a hierarchy of widely separated time scales. Here we carry out the same programme for a discrete-time dynamics in which all Ising spins are updated at once. This dynamics, synchronous Monte Carlo (SyncMC), introduces Gaussian auxiliary fields so that the spins become conditionally independent, and it samples the Boltzmann distribution exactly for any system with symmetric pairwise couplings. We show that, for any such system, the correlation and response functions of SyncMC obey an exact fluctuation-dissipation relation in discrete time in equilibrium. The dynamical mean-field equations of SyncMC for the SK model contain two terms absent in continuous-time dynamics, a self-coupling of each spin and a noise correlated through the couplings; both affect only the fast relaxation. As a result, under the same assumptions as in the Sompolinsky-Zippelius construction (widely separated time scales, and on each slow scale the same relation with a reduced ratio x, as expected in relaxation from random initial conditions), the long-time structure of SyncMC reproduces the Parisi equations, including the Parisi partial differential equation and the equation for the distribution of local fields. Equilibrium simulations with replica exchange for up to N=2048 spins exclude the replica-symmetric distribution of the actual local fields and are consistent with the Parisi solution. The function x(q) estimated from the overlap distribution, and from the response in relaxation from random configurations for up to N=4096 spins, is also consistent with Parisi's.

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Yoshiyuki Kabashima. 2026-10-04. Parisi's solution of the Sherrington--Kirkpatrick model from synchronous Monte Carlo dynamics. https://arxiv.org/abs/2610.05557

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