Directed polymers in heavy-tail random environment

TL;DR

This paper investigates the scaling limits of directed polymers in a heavy-tailed environment, revealing five regimes for fluctuation scales when tail exponent exceeds 1/2, and only two when below, using entropy-controlled last passage percolation.

Contribution

It characterizes all possible fluctuation regimes and scaling limits for directed polymers in heavy-tail environments, confirming a conjecture and introducing the E-LPP method.

Findings

Five regimes of fluctuation scales identified for rom 1/2 to 2.

Only two regimes for rom below 1/2, luctuations are rom or luctuations.

Scaling limits and phase transitions are explicitly characterized.

Abstract

We study the directed polymer model in dimension ${1+1}$ when the environment is heavy-tailed, with a decay exponent $\alpha\in(0,2)$. We give all possible scaling limits of the model in the weak-coupling regime, i.e., when the inverse temperature temperature $\beta=\beta_n$ vanishes as the size of the system $n$ goes to infinity. When $\alpha\in(1/2,2)$, we show that all possible transversal fluctuations $\sqrt{n} \leq h_n \leq n$ can be achieved by tuning properly $\beta_n$, allowing to interpolate between all super-diffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [cf:DZ] - we actually identify five different regimes. On the other hand, when $\alpha<1/2$, we show that there are only two regimes: the transversal fluctuations are either $\sqrt{n}$ or $n$. As a key ingredient, we use the Entropy-controlled Last Passage Percolation (E-LPP), introduced in a companion paper [cf:BT_ELPP].