Delocalization-localization dynamical phase transition of random walks on graphs

TL;DR

This paper investigates a first-order dynamical phase transition in random walks on graphs, revealing a coexistence of localized and delocalized paths, with analytical characterization of finite-size effects and robustness across topologies.

Contribution

It analytically demonstrates a first-order dynamical phase transition in random walks on graphs and characterizes the finite-size crossover and robustness of this transition.

Findings

Identification of a first-order dynamical phase transition in random walks.

Analytical characterization of the finite-size crossover between regimes.

Robustness of the phase transition with respect to graph topology.

Abstract

We consider random walks evolving on two models of connected and undirected graphs and study the exact large deviations of a local dynamical observable. We prove, in the thermodynamic limit, that this observable undergoes a first-order dynamical phase transition (DPT). This is interpreted as a `co-existence' of paths in the fluctuations that visit the highly connected bulk of the graph (delocalization) and paths that visit the boundary (localization). The methods we used also allow us to characterize analytically the scaling function that describes the finite size crossover between the localized and delocalized regimes. Remarkably, we also show that the DPT is robust with respect to a change in the graph topology, which only plays a role in the crossover regime. All results support the view that a first-order DPT may also appear in random walks on infinite-size random graphs.