Exact solutions of the Kardar–Parisi–Zhang equation and weak universality for directed random polymers

Summary

We survey recent results of convergence to random matrix distributions of directed random polymer free energy fluctuations in the intermediate disorder regime. These are obtained by passing through the exact formulas for fluctuations of KPZ at finite time.

1. Directed random polymers

Directed random polymers were introduced in the mid eighties as models of defect lines in media with impurities (see [Kardar 2007] for a review). They became popular in physics because besides their applicability as models and inherent interest, they are a case where the replica methods developed for the more difficult spin glass models give consistent answers. We will be interested in the 1 + 1 dimensional case. We are given a random environment ξ(i, j) of independent identically distributed real random variables for i, j in ℤ₊ × ℤ. Given the environment, the energy of an n-step nearest neighbour walk x =(x₁,..., x_n is

The polymer measure on such walks starting at 0 at time 0 and ending at x at time n is then defined by

The parameter β > 0, which measures how much the path prefers to travel through areas of low energy, is called the inverse temperature. P is the uniform probability measure on such walks, and Z(n, x) is the partition function

This is the point-to-point free energy. If we do not specify the endpoint, we get the point-to-line free energy, which we denote by Z^β,ξ(n).

What happens is that for large n that path is localized about a path which is special for that n; it has lateral fluctuations of size n^⅔. In terms of the free energy, the key conjecture is that its fluctuations are of size n^⅓ and given by

Tracy–Widom distributions.