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

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.

arXiv comment15 pages. Restructured the manuscript

References in corpus (12)

• A KPZ Cocktail- Shaken, not stirred: Toasting 30 years of kinetically roughened surfaces
• Scaling for a one-dimensional directed polymer with boundary conditions
• The one-dimensional KPZ equation and its universality class
• Random-walk in Beta-distributed random environment
• The strict-weak lattice polymer
• Ratios of partition functions for the log-gamma polymer
• KPZ Scaling Theory and the Semi-discrete Directed Polymer Model
• Geometry of geodesics through Busemann measures in directed last-passage percolation
• On integrable directed polymer models on the square lattice
• Busemann functions and Gibbs measures in directed polymer models on ${\mathbb{Z}}^2$
• Large deviations and wandering exponent for random walk in a dynamic beta environment
• Stationary measures for two dual families of finite and zero temperature models of directed polymers on the square lattice

Cited by in corpus (1)

• Dynamic polymers: invariant measures and ordering by noise