The spectral boundary of the Asymmetric Simple Exclusion Process (ASEP) -- free fermions, Bethe ansatz and random matrix theory

TL;DR

This paper analyzes the spectral boundary of the ASEP, revealing spike patterns linked to Bethe roots and random matrix correlations, advancing understanding of non-Hermitian fermionic models in non-equilibrium physics.

Contribution

It introduces a detailed spectral analysis of ASEP, connecting Bethe ansatz, free fermions, and random matrix theory to characterize spectral boundary spikes.

Findings

Spectral boundary exhibits L or L+1 spikes depending on boundary conditions.

Spikes originate from clustering of Bethe roots in the spectrum.

Random matrix models show similar spectral spike patterns.

Abstract

In non-equilibrium statistical mechanics, the Asymmetric Simple Exclusion Process (ASEP) serves as a paradigmatic example. We investigate the spectral characteristics of the ASEP, focusing on the spectral boundary of its generator matrix. We examine finite ASEP chains of length $L$, under periodic (pbc) and open boundary conditions (obc). Notably, the spectral boundary exhibits $L$ spikes for pbc and $L+1$ spikes for obc. Treating the ASEP generator as an interacting non-Hermitian fermionic model, we extend the model to have tunable interaction. In the non-interacting case, the analytically computed many-body spectrum shows a spectral boundary with prominent spikes. For pbc, we use the coordinate Bethe ansatz to interpolate between the noninteracting case to the ASEP limit, and show that these spikes stem from clustering of Bethe roots. The robustness of the spikes in the spectral boundary is demonstrated by linking the ASEP generator to random matrices with trace correlations or, equivalently, random graphs with distinct cycle structures, both displaying similar spiked spectral boundaries.