KPZ scaling theory and the semidiscrete directed polymer model

Summary

We explain how the claims of KPZ scaling theory are confirmed by a recent proof of Borodin and Corwin on the asymptotics of the semidiscrete directed polymer.

1. Introduction

The Kardar–Parisi–Zhang (KPZ) equation [1986] is a stochastic partial differential equation modeling surface growth and, more generally, the motion of an interface bordering a stable against a metastable phase. Scaling theory is an educated guess on the nonuniversal coefficients in the asymptotics for models in the KPZ universality class. Scaling theory has been developed in a landmark contribution by Krug, Meakin and Halpin-Healy [Krug et al. 1992]. The purpose of our article is to explain how to apply scaling theory to the semidiscrete directed polymer. This model has been discussed in depth at the 2010 random matrix workshop at MSRI and, so-to-speak as a spin-off, Borodin and Corwin [2014] developed the beautiful theory of Macdonald processes, which provides the tools for an asymptotic analysis of the semidiscrete directed polymer. As we will establish, scaling theory is consistent with the results of Borodin and Corwin, thereby providing a highly nonobvious control check.

To place the issue in focus, let me start with a simple example. Assume as given the stationary sequence X_j , j ∈ ℤ, of mean zero random variables and let us consider the partial sums

As well studied, it is fairly common that Sn/√n converges to a Gaussian as n→∞, i.e.,

where F_G is the distribution function of a unit Gaussian random variable. Here F_G is the universal object, while the coefficient D > 0 depends on the law ℙ and

is in this sense model dependent, resp. nonuniversal. However, using stationarity, D is readily guessed as

The KPZ class deals with strongly dependent random variables, for which partial sums are of size n^⅓ rather than n^½. F_G is to be substituted by the GUE Tracy–Widom distribution function, F_GUE, which first appeared in the context of the largest eigenvalue of a F_GUE random matrix [Forrester 1993; Tracy and Widom 1994].