Understanding random-walk dynamical phase coexistence through waiting times

TL;DR

This paper investigates how first-order dynamical phase transitions manifest as intermittent co-existing phases in random walks on graphs, highlighting the critical role of waiting times in phase switching and ergodicity.

Contribution

It introduces the concept that diverging waiting times are key to understanding dynamical phase coexistence and critical behavior in random walk fluctuations.

Findings

Waiting times diverge at phase transition points.

Critical behavior is linked to the system's relaxation dynamics.

Ergodicity is maintained at criticality due to diverging timescales.

Abstract

We study the appearance of first-order dynamical phase transitions (DPTs) as `intermittent' co-existing phases in the fluctuations of random walks on graphs. We show that the diverging time scale leading to critical behaviour is the waiting time to jump from one phase to another. This time scale is crucial for observing the system's relaxation to stationarity and demonstrates ergodicity of the system at criticality. We illustrate these results through three analytical examples which provide insights into random walks exploring random graphs.