Directed polymers in random environment with heavy tails

TL;DR

This paper investigates directed polymers in a 1+1 dimensional random environment with heavy-tailed distributions, revealing localization phenomena, limiting measures, and phase transitions depending on tail heaviness and temperature.

Contribution

It introduces a new analysis of directed polymers with polynomially decaying tails, identifying limiting distributions and phase transitions related to tail index and temperature.

Findings

Polymer localizes in favorable environment regions.

Limiting distribution of the polymer's shape identified.

Existence of a critical temperature depending on tail heaviness.

Abstract

We study the model of Directed Polymers in Random Environment in 1+1 dimensions, where the distribution at a site has a tail which decays regularly polynomially with power \alpha, where \alpha \in (0,2). After proper scaling of temperature \beta^{-1}, we show strong localization of the polymer to a favorable region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (\alpha, \beta)-indexed family of measures on Lipschitz curves lying inside the 45-degrees-rotated square with unit diagonal. In particular, this shows order n transversal fluctuations of the polymer. If, and only if, \alpha is small enough, we find that there exists a random critical temperature below which, but not above, the effect of the environment is macroscopic. The results carry over to d+1 dimensions for d>1 with minor modifications.