Theory of stochastic Laplacian growth

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.

21 pages, v2: 20 pages, exposition improved and shortened, 2 figures added

References in corpus (2)

• Normal random matrix ensemble as a growth problem
• Large N expansion for the 2D Dyson gas

Cited by in corpus (5)

• Statistical mechanics of stochastic growth phenomena
• Quantized Laplacian growth, III: On conformal field theories of Laplacian growth
• Two dimensional potential theory with a view towards vortex motion: Energy, capacity and Green functions
• Quantized Laplacian growth, II: 1D hydrodynamics of the Loewner density
• Integrability-preserving regularizations of Laplacian Growth