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September 2015 Ratios of partition functions for the log-gamma polymer
Nicos Georgiou, Firas Rassoul-Agha, Timo Seppäläinen, Atilla Yilmaz
Ann. Probab. 43(5): 2282-2331 (September 2015). DOI: 10.1214/14-AOP933


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.


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Nicos Georgiou. Firas Rassoul-Agha. Timo Seppäläinen. Atilla Yilmaz. "Ratios of partition functions for the log-gamma polymer." Ann. Probab. 43 (5) 2282 - 2331, September 2015.


Received: 1 March 2013; Revised: 1 January 2014; Published: September 2015
First available in Project Euclid: 9 September 2015

zbMATH: 1357.60110
MathSciNet: MR3395462
Digital Object Identifier: 10.1214/14-AOP933

Primary: 60K35 , 60K37

Keywords: 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

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 5 • September 2015
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