Growth processes related to the dispersionless Lax equations

TL;DR

This paper reviews the connection between growth processes like Laplacian growth and slit evolution with integrable systems, specifically dispersionless Lax equations, providing a unified mathematical framework.

Contribution

It introduces a unified approach linking growth processes to dispersionless Lax equations and shows how the L"owner equation emerges as a reduction of the dispersionless Toda hierarchy.

Findings

Lax equations interpret growth as conformal map evolution

L"owner equation arises from hierarchy reductions

Growth processes simulated by Dyson gas models

Abstract

This paper is a short review of the connection between certain types of growth processes and the integrable systems theory, written from the viewpoint of the latter. Starting from the dispersionless Lax equations for the 2D Toda hierarchy, we interpret them as evolution equations for conformal maps in the plane. This provides a unified approach to evolution of smooth domains (such as Laplacian growth) and growth of slits. We show that the L\"owner differential equation for a parametric family of conformal maps of slit domains arises as a consistency condition for reductions of the dispersionless Toda hierarchy. It is also demonstrated how the both types of growth processes can be simulated by the large $N$ limit of the Dyson gas picture for the model of normal random matrices.