Stochastically growing interfaces, KPZ universality and the weakly asymmetric exclusion process

2011. január 27. csütörtök 16:15

Stochastically growing interfaces play an important role in physics and mathematics. There exist a variety of models which have been created to simulate the behaviour of interfaces growing stochastically according to local rules. Depending on the nature of these rules, the interfaces fluctuate around their limiting profile in different ways.

In 1986 Kardar-Parisi-Zhang (KPZ) predicted that a large class of these models would share the same order of fluctuations. Roughly put, a model is said to be in the KPZ universality class if the fluctuations of it's height function are of order T^1/3 (compare this to the N^1/2 fluctuation of CLT or the fluctuations of the largest eigenvalue in random matrix theory). It is believed that a wide variety of models, including directed polymers, are part of this universality class. Underlying this prediction was a continuum object - the stochastic PDE now known as the KPZ equation, which is supposed to govern the dynamics of models in this class.

After introducing some ideas concerning growth models and their universality classes, we will focus on a continuum model (the directed continuous random polymer) which is related to both the KPZ equation and the stochastic heat equation. We use a suitable discrete particle system (namely "weakly" asymmetric exclusion, which will be defined in the lecture) to solve the exact probability distribution for the free energy of the continuum random polymer. Our distribution also gives the one-point marginals of the stochastic heat equation with delta initial data, and of the solution to the KPZ equation with "narrow wedge" initial data. We also show how these distributions constitute a crossover between the KPZ and the EW class which exhibits Gaussian scale fluctuations (n^1/2).

No prior knowledge of growth processes, exclusion processes or the KPZ equation will be assumed.