The high-temperature behavior for the directed polymer in dimension 1+2

TL;DR

This paper analyzes the high-temperature behavior of the directed polymer model in 1+2 dimensions, revealing the asymptotic decay of the difference between quenched and annealed free energies as temperature increases.

Contribution

The authors adapt techniques from disordered pinning models to precisely characterize the asymptotic behavior of free energy differences in the directed polymer model at high temperatures.

Findings

elta F(eta) decays exponentially as eta o 0

symptotic behavior given by eta^2 \, \log \Delta F(eta) o -\pi

stablishes a sharp asymptotic formula for high-temperature limit

Abstract

We investigate the high-temperature behavior of the directed polymer model in dimension $1+2$. More precisely we study the difference $\Delta \mathtt{F}(\beta)$ between the quenched and annealed free energies for small values of the inverse temperature $\beta$. This quantity is associated to localization properties of the polymer trajectories, and is related to the overlap fraction of two replicas. Adapting recent techniques developed by the authors in the context of the disordered pinning model (Berger and Lacoin, arXiv:1503.07315 [math-ph]), we identify the sharp asymptotic high temperature behavior \[\lim_{\beta\to 0} \, \beta^2 \log \Delta \mathtt{F}(\beta) = -\pi \, .\]