Free energy of directed polymers in random environment in $1+1$-dimension at high temperature

TL;DR

This paper investigates the high-temperature behavior of the free energy of directed polymers in a 1+1-dimensional random environment, establishing a precise limit related to the stochastic heat equation under certain potential conditions.

Contribution

It proves the exact asymptotic limit of the free energy as temperature increases, linking it to the solution of the stochastic heat equation with white noise.

Findings

Limit of free energy normalized by eta^4 is -1/6 as eta approaches zero.

The result connects polymer free energy to the stochastic heat equation.

Provides conditions under which the asymptotic behavior holds.

Abstract

We consider the free energy $F(\beta)$ of the directed polymers in random environment in $1+1$-dimension. It is known that $F(\beta)$ is of order $-\beta^4$ as $\beta\to 0$. In this paper, we will prove that under a certain condition of the potential, \begin{align*} \lim_{\beta\to 0}\frac{F(\beta)}{\beta^4}=\lim_{T\to\infty}\frac{1}{T}P_{\mathcal{Z}}\left[\log \mathcal{Z}_{\sqrt{2}}(T)\right] =-\frac{1}{6}, \end{align*} where $\{\mathcal{Z}_\beta(t,x):t\geq 0,x\in\mathbb{R}\}$ is the unique mild solution to the stochastic heat equation \begin{align*} \frac{\partial}{\partial t}\mathcal{Z}=\frac{1}{2}\Delta \mathcal{Z}+\beta \mathcal{Z}{\dot{\mathcal W}},\ \ \lim_{t\to 0}\mathcal{Z}(t,x)dx=\delta_{0}(dx), \end{align*} where $\mathcal{W}$ is a time-space white noise and \begin{align*} \mathcal{Z}_\beta(t)=\int_\mathbb{R}\mathcal{Z}_\beta(t,x)dx. \end{align*}