Correlation thresholds in the steady states of particle systems and spin glasses

TL;DR

This paper investigates how the steady-state distributions of particle systems and spin glasses relate to an effective potential, revealing thresholds where their correlation with exit rates changes dramatically.

Contribution

It provides the first explicit estimates of the correlation between effective potential and exit rates in particle systems and spin glasses, explaining threshold phenomena.

Findings

Correlation determines approximation quality of steady states

Thresholds in correlation relate to transitions in potential structure

Explicit estimates of correlation in specific models

Abstract

A growing body of theoretical and empirical evidence shows that the global steady-state distributions of many equilibrium and nonequilibrium systems approximately satisfy an analogue of the Boltzmann distribution, with a local dynamical property of states playing the role of energy. The correlation between the effective potential of the steady-state distribution and the logarithm of the exit rates determines the quality of this approximation. We demonstrate and explain this phenomenon in a simple one-dimensional particle system and in random dynamics of the Sherrington-Kirkpatrick spin glass by providing the first explicit estimates of this correlation. We find that, as parameters of the dynamics vary, each system exhibits a threshold above and below which the correlation dramatically differs. We explain how these thresholds arise from underlying transitions in the relationship between the local and global "parts" of the effective potential.