The Annals of Probability

Ratios of partition functions for the log-gamma polymer

Nicos Georgiou, Firas Rassoul-Agha, Timo Seppäläinen, and Atilla Yilmaz

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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.

Article information

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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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments

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.

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