Stationary inverse-Wishart polymers

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.