Statistical mechanics of stochastic growth phenomena

TL;DR

This paper develops a statistical mechanics framework for stochastic growth phenomena, linking Laplacian growth with random matrix theory and thermodynamic fluctuations to describe domain transitions.

Contribution

It introduces a novel connection between Laplacian growth, random matrix theory, and thermodynamic fluctuations, providing a new stochastic growth equation and transition probability framework.

Findings

Laplacian growth derived from a variational principle.

Transitions described by a stochastic Laplacian growth equation.

Transition probabilities match the free-particle propagator on a complex manifold.

Abstract

We develop statistical mechanics for stochastic growth processes as applied to Laplacian growth by using its remarkable connection with a random matrix theory. The Laplacian growth equation is obtained from the variation principle and describes adiabatic (quasi-static) thermodynamic processes in the two-dimensional Dyson gas. By using Einstein's theory of thermodynamic fluctuations we consider transitional probabilities between thermodynamic states, which are in a one-to-one correspondence with planar domains. Transitions between these domains are described by the stochastic Laplacian growth equation, while the transitional probabilities coincide with the free-particle propagator on the infinite dimensional complex manifold with the K\"ahler metric.