Stationary inverse-Wishart polymers

• Zikun Ouyang (Université Paris Dauphine)

A solvable model of directed polymer with matrix-valued disorder is introduced by Arista, Bisi and O'Connell 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 talk, 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. Based on joint work with G. Barraquand, arXiv:2511.14375.

The formation of a soliton gas condensate for the focusing Nonlinear Schrödinger equation.

• Aikaterini Gkogkou (Tulane University)

In this talk, we consider the focusing Nonlinear Schrödinger (NLS) equation and its multi-soliton solution when the number of solitons grows to infinity. We discover configurations of multi-soliton solutions that exhibit the formation of a soliton gas condensate. Specifically, we show that when the associated discrete eigenvalues accumulate on two bounded horizontal segments in the complex plane, and the associated norming constants are bounded away from zero, the solution to the focusing NLS equation is described by a rapidly oscillatory elliptic wave with constant velocity, on compact subsets of the $(x,t)$ domain.