Stationary inverse-Wishart polymers

TL;DR

This paper introduces stationary inverse-Wishart polymer models on a lattice, generalizing scalar models to matrix-valued disorder, and characterizes their stationary measures and conjectures about their asymptotic free energy behavior.

Contribution

It extends directed polymer models to matrix-valued inverse-Wishart disorder, providing explicit stationary measures and exploring differences from scalar models.

Findings

Explicit stationary measures characterized for the models

Connection to random walks with inverse-Wishart increments in special cases

Conjectures on asymptotic free energy behavior

Abstract

A solvable model of directed polymer with matrix-valued disorder is introduced in arXiv:2203.14868. The disorder is made of $d\times d$ inverse-Wishart random matrices, so that the model nicely generalizes the well-studied log-gamma polymer, recovered when $d=1$. Much of the features of the log-gamma polymer seem to have analogues for higher $d$, although the integrability needs to be better understood. In this paper, we introduce stationary inverse-Wishart polymer models on a quadrant or a strip of $\mathbb Z^2$. In each setting, we identify stationary measures, characterized explicitly in terms of random walks with inverse-Wishart increments in special cases, or more complicated two-layer Gibbs measures for generic choices of boundary parameters. We also make conjectures about asymptotics of the free energy, and explain important differences between matrix-valued polymer models and their scalar counterpart, due to non-commutativity.