# Uniqueness and ergodicity of stationary directed polymers on $\mathbb Z^2$

## TL;DR

This paper investigates the uniqueness and ergodic properties of stationary directed polymer models on the two-dimensional integer lattice with i.i.d. weights, establishing conditions for ergodicity and uniqueness of the stationary distribution.

## Contribution

It proves the uniqueness of ergodic stationary polymer distributions given weight distribution and corrector mean, and characterizes ergodicity under certain moment and extremality conditions.

## Key findings

- At most one ergodic stationary polymer distribution exists for given weights and corrector mean.
- Ergodicity under both shifts is guaranteed if weights have more than two moments and the corrector mean is extremal.
- Results contribute to understanding the ergodic structure of directed polymers on $\

## Abstract

We study the ergodic theory of stationary directed nearest-neighbor polymer models on $\mathbb Z^2$, with i.i.d. weights. Such models are equivalent to specifying a stationary distribution on the space of weights and correctors that satisfy certain consistency conditions. We show that for prescribed weight distribution and corrector mean, there is at most one stationary polymer distribution which is ergodic under the $e_1$ or $e_2$ shift. Further, if the weights have more than two moments and the corrector mean vector is an extreme point of the superdifferential of the limiting free energy, then the corrector distribution is ergodic under each of the $e_1$ and $e_2$ shifts.