Asymmetric simple exclusion process on a ring conditioned on enhanced flux

TL;DR

This paper analyzes the ASEP on a ring conditioned on high flux, revealing an effective long-range potential akin to random matrix eigenvalue interactions, and derives related probabilities using advanced analytical techniques.

Contribution

It introduces a novel effective potential for ASEP under flux conditioning and applies Bethe ansatz and determinantal methods to derive analytical results.

Findings

Effective potential resembles eigenvalue interactions in random matrices.

Analytical expressions for hopping rates and quasistationary probabilities.

Results extend to large current and activity in related reaction-diffusion processes.

Abstract

We show that in the asymmetric simple exclusion process (ASEP) on a ring, conditioned on carrying a large flux, the particle experience an effective long-range potential which in the limit of very large flux takes the simple form $U= -2\sum_{i\neq j}\log|\sin\pi(n_{i}/L-n_{j}/L)|$, where $n_{1}% n_{2},\ldots n_{N}$ are the particle positions, similar to the effective potential between the eigenvalues of the circular unitary ensemble in random matrices. Effective hopping rates and various quasistationary probabilities under such a conditioning are found analytically using the Bethe ansatz and determinantal free fermion techniques. Our asymptotic results extend to the limit of large current and large activity for a family of reaction-diffusion processes with on-site exclusion between particles. We point out an intriguing generic relation between classical stationary probability distributions for conditioned dynamics and quantum ground state wave functions, in particular, in the case of exclusion processes, for free fermions.