A strange five vertex model and multispecies ASEP on a ring

TL;DR

This paper introduces a novel five vertex model with unique weight conservation to analyze the stationary states of multispecies ASEP on a ring, clarifying and connecting various existing results through diagrammatic and algebraic methods.

Contribution

It presents a new five vertex model with a strange weight conservation, linking multiple known approaches to multispecies ASEP and providing new diagrammatic and algebraic insights.

Findings

Identifies stationary probability as a partition function of a 3D system.

Provides a diagrammatic interpretation of matrix product operators as corner transfer matrices.

Reveals the origin of recursive tensor structures via CTM diagrams.

Abstract

We revisit the problem of constructing the stationary states of the multispecies asymmetric simple exclusion process on a one-dimensional periodic lattice. Central to our approach is a quantum oscillator weighted five vertex model which features a strange weight conservation distinct from the conventional one. Our results clarify the interrelations among several known results and refine their derivations. For instance, the stationary probability derived from the multiline queue construction by Martin (2020) and Corteel--Mandelshtam--Williams (2022) is identified with the partition function of a three-dimensional system. The matrix product operators by Prolhac--Evans--Mallick (2009) acquire a natural diagrammatic interpretation as corner transfer matrices (CTM). The origin of their recursive tensor structure, as questioned by Aggarwal--Nicoletti--Petrov (2023), is revealed through the CTM diagrams. Finally, the derivation of the Zamolodchikov--Faddeev algebra by Cantini--de Gier--Wheeler (2015) is made intrinsic by elucidating its precise connection to a solution to the Yang--Baxter equation originating from quantum group representations.