Free Energy of the Cauchy Directed Polymer Model at High Temperature

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*}