Another Look at the KPZ Problem

We will start by introducing the phenomenon of the KPZ (Kardar-Parisi-Zhang) universalty. KPZ problem was a very active research area in the last 20 years. The problem is essentially interdisciplinary. It is related to such fields as probability theory, statistical mechanics, mathematical physics, PDE, SPDE, random dynamics, random matrices, and random geometry, to name a few.  In the most general form the problem can be formulated in the following way. Consider random geometry on the two-dimensional plane. We shall think about it as a random landscape of hills, mountains, and valleys. The main aim is to understand the asymptotic statistical properties of the length of the geodesic connecting two points in the limit as distance between the endpoints tends to infinity.  One also wants to study the geometry of random geodesics, in particular how much they deviate from a straight line.  It turns out that the limiting statistics for both the length and the deviation is universal, that is it does not depend on the probability distribution of the random landscape. Moreover, many limiting probability distributions can be found explicitly. Thus, KPZ phenomenon exhibits both universality and integrability.

 

In the second part of the talk I plan to introduce an alternative geometrical approach to the problem of the KPZ universality.  It provides an even broader point of view on the problem of universal statistical behavior.