# Directed polymer in $\gamma$-stable Random Environments

## TL;DR

This paper investigates how the phase transition in directed polymers in random environments is affected by heavy-tailed disorder, replacing the traditional exponential model with a stable law domain of attraction.

## Contribution

It introduces a model with heavy-tailed disorder variables in directed polymers and explores the impact on phase transition behavior across dimensions.

## Key findings

- Heavy-tailed disorder alters the existence of phase transition.
- The model extends understanding beyond finite second moment assumptions.
- Results depend on the tail index  in the stable law domain.

## Abstract

The transition from a weak-disorder (diffusive phase) to a strong-disorder (localized phase) for directed polymers in a random environment is a well studied phenomenon. In the most common setup, it is established that the phase transition is trivial when the transversal dimension $d$ equals $1$ or $2$ (the diffusive phase is reduced to $\beta=0$) while when $d\geq 3$, there is a critical temperature $\beta_c\in (0,\infty)$ which delimits the two phases. The proof of the existence of a diffusive regime for $d\geq 3$ is based on a second moment method, and thus relies heavily on the assumption that the variable which encodes the disorder intensity (which in most of the mathematics literature assumes the form $e^{\beta \eta_x}$), has finite second moment. The aim of this work is to investigate how the presence/absence of phase transition may depend on the dimension $d$ in the case when the disorder variable displays heavier tail. To this end we replace $e^{\beta \eta_x}$ by $(1+\beta \omega_x)$ where $\omega_x$ is in the domain of attraction of a stable law with parameter $\gamma \in (1, 2)$.