Rate of convergence for polymers in a weak disorder

TL;DR

This paper analyzes the convergence rate of the normalized partition function for directed polymers in a weak disorder regime on a lattice, showing it converges to a Gaussian distribution with specific scaling properties.

Contribution

It establishes the polynomial rate of convergence and the Gaussian limit distribution for the normalized partition function in weak disorder on Z^d, highlighting differences from tree models.

Findings

Normalized partition function converges to a Gaussian law.

Convergence rate scales as n(d-2)/4.

Distinct behavior from polymers on trees.

Abstract

We consider directed polymers in random environment on the lattice Z d at small inverse temperature and dimension d $\ge$ 3. Then, the normalized partition function W n is a regular martingale with limit W. We prove that n (d--2)/4 (W n -- W)/W n converges in distribution to a Gaussian law. Both the polynomial rate of convergence and the scaling with the martingale W n are different from those for polymers on trees.