Tail decay for the distribution of the endpoint of a directed polymer

TL;DR

This paper derives an asymptotic tail expansion for the distribution of the endpoint of a directed polymer, connecting Airy processes and Painlevé equations to quantify large deviations.

Contribution

It provides the first detailed asymptotic expansion for the tail distribution of the polymer endpoint using advanced integrable probability techniques.

Findings

Tail probability decays as $Ce^{-4/3\varphi(t)}t^{-145/32}$ for large $t$

Explicit formula for the constant $C$ in the tail asymptotics

Connection established between Airy$_2$ process, Painlevé II, and polymer endpoint distribution

Abstract

We obtain an asymptotic expansion for the tails of the random variable $\tcal=\arg\max_{u\in\mathbb{R}}(\mathcal{A}_2(u)-u^2)$ where $\mathcal{A}_2$ is the Airy$_2$ process. Using the formula of Schehr \cite{Sch} that connects the density function of $\tcal$ to the Hastings-McLeod solution of the second Painlev\'e equation, we prove that as $t\rightarrow\infty$, $\mathbb{P}(|\tcal|>t)=Ce^{-4/3\varphi(t)}t^{-145/32}(1+O(t^{-3/4}))$, where $\varphi(t)=t^3-2t^{3/2}+3t^{3/4}$, and the constant $C$ is given explicitly.