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,

$$\begin{aligned} \lim \limits _{\beta \rightarrow 0}\beta ^{2}\log (-p(\beta ))=-c. \end{aligned}$$