Bulk-Boundary Correspondence in Ergodic and Nonergodic One-Dimensional Stochastic Processes

TL;DR

This paper establishes a general bulk-boundary correspondence principle for classical stochastic processes, linking topological invariants to localized steady states, applicable to both ergodic and nonergodic systems, including many-body models like ASEP.

Contribution

It proves a universal bulk-boundary correspondence in stochastic processes and extends it to complex many-body models, advancing topological understanding in non-equilibrium systems.

Findings

Winding number correlates with localized steady states.

The correspondence applies to both ergodic and nonergodic systems.

Extended to many-body stochastic models like ASEP.

Abstract

Bulk-boundary correspondence is a fundamental principle in topological physics. In recent years, there have been considerable efforts in extending the idea of geometry and topology to classical stochastic systems far from equilibrium. However, it has been unknown whether or not the bulk-boundary correspondence can be extended to the steady states of stochastic processes accompanied by additional constraints such as the conservation of probability. The present study reveals the general form of bulk-boundary correspondence in classical stochastic processes. Specifically, we prove a correspondence between the winding number and the number of localized steady states in both ergodic and nonergodic systems. Furthermore, we extend the argument of the bulk-boundary correspondence to a many-body stochastic model called the asymmetric simple exclusion process (ASEP). These results would provide a guiding principle for exploring topological origin of localization in various stochastic processes including biological systems.