Dynamical Freezing in a Spin Glass System with Logarithmic Correlations

TL;DR

This paper rigorously demonstrates dynamical freezing in a spin glass system with logarithmic correlations by analyzing a random walk influenced by a Gaussian free field, revealing a scaling limit as a supercritical Liouville Brownian motion.

Contribution

It provides the first rigorous proof of dynamical freezing in a spin glass with logarithmic correlations, linking the process to a supercritical Liouville Brownian motion.

Findings

Scaling limit as a spatial K-process with a random trapping landscape

Explicit relation to the extremal process of the Gaussian free field

Demonstration of dynamical freezing in a logarithmically correlated spin glass

Abstract

We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. Alternatively, the limiting process is a supercritical Liouville Brownian motion with respect to the continuum Gaussian free field on the box. This demonstrates rigorously and for the first time, as far as we know, a dynamical freezing in a spin glass system with logarithmically correlated energy levels.