2013/03/31 by Nicos Georgiou, Firas Rassoul‐Agha, Firas Rassoul-Agha +2 · 2 citations
Mathematics · #Combinatorics #Duality (order theory) #Ergodic theory #Limit (mathematics) #Limiting #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematics #Partition (number theory) #Partition function (quantum field theory) #Physics #Polymer #Quantum mechanics #Random Matrices and Applications #Random walk #Statistical physics #Statistics #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.1214/14-aop933
published as Annals of Probability 2015, Vol. 43, No. 5, 2282-2331 · Published at http://dx.doi.org/10.1214/14-AOP933 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2015/09/01 · arxiv created 2015/10/28 · arxiv updated 2015/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.