Subgaussian concentration and rates of convergence in directed polymers

TL;DR

This paper establishes subgaussian concentration results for the partition function of directed polymers in nearly gamma disorder, providing new insights into the convergence rates of the free energy in high dimensions.

Contribution

It introduces subgaussian concentration bounds for the log-partition function and quantifies the rate at which the expected log-partition function converges to the free energy.

Findings

Exponential concentration of log Z around its mean on the scale √(N/ log N)

The difference between expected log Z and N times free energy is subgaussian, specifically O(√(N/ log N) log log N)

Provides rigorous bounds on convergence rates in directed polymer models with nearly gamma disorder.

Abstract

We consider directed random polymers in $(d+1)$ dimensions with nearly gamma i.i.d. disorder. We study the partition function $Z_{N,\omega}$ and establish exponential concentration of $\log Z_{N,\omega}$ about its mean on the subgaussian scale $\sqrt{N/\log N}$ . This is used to show that $\mathbb{E}[ \log Z_{N,\omega}]$ differs from $N$ times the free energy by an amount which is also subgaussian (i.e. $o(\sqrt{N})$), specifically $O(\sqrt{\frac{N}{\log N}}\log \log N)$.