Ratios of partition functions for the log-gamma polymer

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.

Published at http://dx.doi.org/10.1214/14-AOP933 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Distribution functions for largest eigenvalues and their applications
• Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
• Competition interfaces and second class particles
• A phase transition for competition interfaces
• Stochastic integrability and the KPZ equation
• Busemann functions and infinite geodesics in two-dimensional first-passage percolation

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