Limiting results for the free energy of directed polymers in random environment with unbounded jumps

TL;DR

This paper investigates the asymptotic behavior of the free energy for directed polymers with unbounded jumps in a Bernoulli environment, establishing existence, continuity, and asymptotics at extreme coupling constants.

Contribution

It introduces a novel proof of free energy existence at , avoiding subadditivity, and characterizes its asymptotics using a directed first passage percolation time constant.

Findings

Existence and continuity of free energy including at 

Asymptotic description of free energy at  in Bernoulli environment

Continuity property of the time constant of directed first passage percolation

Abstract

We study asymptotics of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by Bernoulli variables. We first establish the existence and continuity of the free energy including the negative infinity value of the coupling constant $\beta$. Our proof of existence at $\beta=-\infty$ differs from existing ones in that it avoids the direct use of subadditivity. Secondly, we identify the asymptotics of the free energy at $\beta=-\infty$ in the limit of the success probability of the Bernoulli variables tending to one. It is described by using the so-called time constant of a certain directed first passage percolation. Our proof relies on a certain continuity property of the time constant, which is of independent interest.