Dynamical phase transition to localized states in the two-dimensional random walk conditioned on partial currents

TL;DR

This paper investigates a dynamical phase transition in a two-dimensional random walk conditioned on partial currents, revealing a shift from delocalized to localized states and spontaneous symmetry breaking, with implications for non-equilibrium physics.

Contribution

It provides the first microscopic and analytical characterization of a dynamical phase transition in a non-interacting 2D random walk conditioned on partial currents.

Findings

Identification of a phase transition between delocalized and localized states.

Spontaneous Z2-symmetry breaking accompanies the transition.

Numerical and analytical methods elucidate the phases and transition mechanism.

Abstract

The study of dynamical large deviations allows for a characterization of stationary states of lattice gas models out of equilibrium conditioned on averages of dynamical observables. The application of this framework to the two-dimensional random walk conditioned on partial currents reveals the existence of a dynamical phase transition between delocalized band dynamics and localized vortex dynamics. We present a numerical microscopic characterization of the phases involved, and provide analytical insight based on the macroscopic fluctuation theory. A spectral analysis of the microscopic generator shows that the continuous phase transition is accompanied by spontaneous $\mathbb{Z}_2$-symmetry breaking whereby the stationary solution loses the reflection symmetry of the generator. Dynamical phase transitions similar to this one, which do not rely on exclusion effects or interactions, are likely to be observed in more complex non-equilibrium physics models.