Theory of stochastic Laplacian growth

TL;DR

This paper extends diffusion-limited aggregation to multiple particles, linking growth probabilities to integrable systems, Hamiltonian dynamics, and conformal field theory, revealing deep mathematical structures in stochastic Laplacian growth.

Contribution

It introduces a probabilistic framework for stochastic Laplacian growth, connecting it with integrable hierarchies, Hamiltonian structures, and conformal field theory, and identifies the most probable growth path as deterministic Laplacian growth.

Findings

Most probable growth follows deterministic Laplacian growth.

Growth probabilities relate to tau-functions of integrable hierarchies.

Established Hamiltonian structure for interface dynamics.

Abstract

We generalize the diffusion-limited aggregation by issuing many randomly-walking particles, which stick to a cluster at the discrete time unit providing its growth. Using simple combinatorial arguments we determine probabilities of different growth scenarios and prove that the most probable evolution is governed by the deterministic Laplacian growth equation. A potential-theoretical analysis of the growth probabilities reveals connections with the tau-function of the integrable dispersionless limit of the two-dimensional Toda hierarchy, normal matrix ensembles, and the two-dimensional Dyson gas confined in a non-uniform magnetic field. We introduce the time-dependent Hamiltonian, which generates transitions between different classes of equivalence of closed curves, and prove the Hamiltonian structure of the interface dynamics. Finally, we propose a relation between probabilities of growth scenarios and the semi-classical limit of certain correlation functions of "light" exponential operators in the Liouville conformal field theory on a pseudosphere.