The Annals of Probability
- Ann. Probab.
- Volume 43, Number 5 (2015), 2282-2331.
Ratios of partition functions for the log-gamma polymer
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.
Ann. Probab. Volume 43, Number 5 (2015), 2282-2331.
Received: March 2013
Revised: January 2014
First available in Project Euclid: 9 September 2015
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Busemann function competition interface convex duality directed polymer geodesic Kardar–Parisi–Zhang universality large deviations log-gamma polymer random environment random walk in random environment variational formula
Georgiou, Nicos; Rassoul-Agha, Firas; Seppäläinen, Timo; Yilmaz, Atilla. Ratios of partition functions for the log-gamma polymer. Ann. Probab. 43 (2015), no. 5, 2282--2331. doi:10.1214/14-AOP933. https://projecteuclid.org/euclid.aop/1441792286.