Closure of the Monte Carlo dynamical equations in the spherical Sherrington-Kirkpatrick model

TL;DR

This paper analytically solves the Monte Carlo dynamics of the spherical Sherrington-Kirkpatrick model, revealing the role of the acceptance rate and simplifying the relaxational behavior at zero temperature.

Contribution

It provides explicit solutions for observables and highlights the acceptance rate as a key factor in the model's dynamics, using generating function techniques.

Findings

Explicit solutions for energy, correlation, and response functions.

Acceptance rate governs the dynamics.

Zero-temperature relaxation resembles a harmonic oscillator.

Abstract

We study the analytical solution of the Monte Carlo dynamics in the spherical Sherrington-Kirkpatrick model using the technique of the generating function. Explicit solutions for one-time observables (like the energy) and two-time observables (like the correlation and response function) are obtained. We show that the crucial quantity which governs the dynamics is the acceptance rate. At zero temperature, an adiabatic approximation reveals that the relaxational behavior of the model corresponds to that of a single harmonic oscillator with an effective renormalized mass.