Ratios of partition functions for the log-gamma polymer

• Georgiou, Nicos ^[1] ; Rassoul-Agha, Firas ^[2] ; Seppäläinen, Timo ^[3] ; Yilmaz, Atilla ^[4]
  1. [1] University of Sussex

     University of Sussex

     Reino Unido

     
  2. [2] University of Utah

     University of Utah

     Estados Unidos

     
  3. [3] University of Wisconsin–Madison

     University of Wisconsin–Madison

     City of Madison, Estados Unidos

     
  4. [4] Boğaziçi University

     Boğaziçi University

     Turquía

     

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 5, 2015, págs. 2282-2331
• Idioma: inglés
• DOI: 10.1214/14-AOP933
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• Resumen

  • 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.