# Free Energy of the Cauchy Directed Polymer Model at High Temperature

## TL;DR

This paper analyzes the free energy behavior of the Cauchy directed polymer model in 1+1 dimensions at high temperatures, establishing negativity and precise asymptotics under certain conditions.

## Contribution

It provides the first rigorous results on the free energy's negativity and asymptotic behavior for the Cauchy directed polymer model at high temperature.

## Key findings

- Free energy is strictly negative for all positive inverse temperatures.
- Sharp asymptotics of free energy as temperature increases are identified.
- The asymptotic limit of β² log(-p(β)) equals -c as β approaches zero.

## Abstract

We study the Cauchy directed polymer model on $\mathbb{Z}^{1+1}$, where the underlying random walk is in the domain of attraction to the $1$-stable law. We show that, if the random walk satisfies certain regularity assumptions and its symmetrized version is recurrent, then the free energy is strictly negative at any inverse temperature $\beta>0$. Moreover, under additional regularity assumptions on the random walk, we can identify the sharp asymptotics of the free energy in the high temperature limit, namely, \begin{equation*} \lim\limits_{\beta\to0}\beta^{2}\log(-p(\beta))=-c. \end{equation*}