Ratios of partition functions for the log-gamma polymer

TL;DR

This paper studies the log-gamma polymer model in 1+1 dimensions, establishing limits of partition function ratios, which lead to Busemann functions, cocycles, and insights into the model's free energy and duality properties.

Contribution

It introduces a new random walk in random environment linked to the log-gamma polymer and proves the existence of limits of partition function ratios, revealing new structural properties.

Findings

Limits of ratios of partition functions are established.

Busemann functions and cocycles are derived from these limits.

A family of ergodic invariant distributions for the associated random walk is identified.

Abstract

We introduce a random walk in random environment associated to an underlying directed polymer model in $1+1$ dimensions. This walk is the positive temperature counterpart of the competition interface of percolation and arises as the limit of quenched polymer measures. We prove this limit for the exactly solvable log-gamma polymer, as a consequence of almost sure limits of ratios of partition functions. These limits of ratios give the Busemann functions of the log-gamma polymer, and furnish centered cocycles that solve a variational formula for the limiting free energy. Limits of ratios of point-to-point and point-to-line partition functions manifest a duality between tilt and velocity that comes from quenched large deviations under polymer measures. In the log-gamma case, we identify a family of ergodic invariant distributions for the random walk in random environment.