The discrete-time parabolic Anderson model with heavy-tailed potential

TL;DR

This paper analyzes a discrete-time parabolic Anderson model with heavy-tailed random potential, revealing almost sure localization of the polymer at a single point with explicit characterization of the localization and paths.

Contribution

It introduces a discrete-time model with heavy-tailed potential and characterizes the almost sure localization and path behavior, extending understanding of polymer models with heavy tails.

Findings

Polymer localizes at a single point almost surely

Localization point grows ballistically with system size

Explicit description of localization and typical paths

Abstract

We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed (1+d)-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the d orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically. We give an explicit characterization of the localization point and of the typical paths of the model.