Free Energy of the Cauchy Directed Polymer Model at High Temperature

We study the Cauchy directed polymer model on 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 β> 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, limβ→0β2log(-p(β))=-c.