The tail distribution of the partition function for directed polymer in the weak disorder phase

TL;DR

This paper analyzes the tail distribution of the partition function for directed polymers in a random environment within the weak disorder phase, revealing power-law decay and extending previous bounded-environment results.

Contribution

It establishes the power-law decay of the partition function's distribution and extends recent results to unbounded environments using new technical estimates.

Findings

Partition function exhibits power-law tail decay with exponent p*(β)

Distribution of supremum of point-to-point and point-to-line partition functions share the same tail behavior

Technical estimate relates L^p-norm of partition function to high-value overshoot

Abstract

We investigate the upper tail distribution of the partition function of the directed polymer in a random environment on $\mathbb Z^d$ in the weak disorder phase. We show that the distribution of the infinite volume partition function $W^{\beta}_{\infty}$ displays a power-law decay, with an exponent $p^*(\beta)\in [1+\frac{2}{d},\infty)$. We also prove that the distribution of the suprema of the point-to-point and point-to-line partition functions display the same behavior. On the way to these results, we prove a technical estimate of independent interest: the $L^p$-norm of the partition function at the time when it overshoots a high value $A$ is comparable to $A$. We use this estimate to extend the validity of many recent results that were proved under the assumption that the environment is upper bounded.