High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder

TL;DR

This paper investigates the universality and limiting behaviors of the directed polymer model in (1+1) dimensions with heavy-tailed disorder, extending known results beyond exponential moments.

Contribution

It proves the universality conjecture for six moments and identifies new universal limits for fewer moments, analyzing the impact on the scaling exponent.

Findings

Universality holds under six moments for the directed polymer model.

Different universal limiting behaviors occur with fewer than six moments.

The scaling exponent for the log-partition function varies with moment assumptions.

Abstract

The directed polymer model at intermediate disorder regime was introduced by Alberts-Khanin-Quastel~\cite{AKQ12}. It was proved that at inverse temperature $\beta n^{-\gamma}$ with $\gamma=1/4$ the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments. We also explain the behavior of the scaling exponent for the log-partition function under different moment assumptions and values of $\gamma$.