2012/02/12 by Firas Rassoul‐Agha, Firas Rassoul-Agha, Rassoul-Agha, Firas +2 · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #60F10 #60K35 #60K37 #82B41 #82D60 #Diffusion and Search Dynamics #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60F10 #msc:60K35 #msc:60K37 #msc:82B41 #msc:82D60
paper · pdf · doi:10.48550/arxiv.1202.2584
39 pages, 3 figures, minor typos fixed
openalex publication_date 2012/02/12 · arxiv created 2013/02/11 · arxiv updated 2013/02/12 · openalex created_date 2022/08/16 · openalex updated_date 2026/07/28
We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and some steps of the walk. The potential can be unbounded, but it is subject to a moment assumption whose strictness is tied to the mixing of the environment, the best case being the i.i.d. environment. We prove that the infinite volume quenched point-to-point free energy exists and has a variational formula in terms of an entropy. We establish regularity properties of the point-to-point free energy, as a function of the potential and as a function on the convex hull of the admissible steps of the walk, and link it to the infinite volume free energy and quenched large deviations of the endpoint of the walk. One corollary is a quenched large deviation principle for random walk in an ergodic random environment, with a continuous rate function.