Quenched point-to-point free energy for random walks in random potentials

• Published in: Probability theory and related fields Vol. 158; no. 3-4; pp. 711 - 750
• Main Authors: Rassoul-Agha, Firas, Seppaelaeinen, Timo
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2014; Springer Nature B.V
• Subjects: Algebra; Continuity (mathematics); Deviation; Economics; Energy; Entropy; Ergodic processes; Finance; Free energy; Insurance; Lattices; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Phase transitions; Polymers; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Quenching; Quenching (cooling); Random walk; Random walk theory; Regularity; Statistics for Business; Studies; Theorems; Theoretical; 60F10; RWRP; Random walk; Point-to-point; 82D60; Polymer; 82B41; Entropy; Free energy; Large deviation; Random potential; Random environment; Stretched polymer; Directed polymer; 60K37; Variational formula; 60K35; RWRE; Quenched
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-013-0494-z

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 steps of the walk. The potential 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 entropy. We establish regularity properties of the point-to-point free energy, and link it to the infinite volume point-to-line free energy and quenched large deviations of the walk. One corollary is a quenched large deviation principle for random walk in an ergodic random environment, with a continuous rate function.