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Ratios of partition functions for the log-gamma polymer. (English) Zbl 1357.60110

Summary: 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
60G50 Sums of independent random variables; random walks
60F10 Large deviations
82D60 Statistical mechanics of polymers
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