TY - JOUR

T1 - Theory of Stochastic Laplacian Growth

AU - Alekseev, Oleg

AU - Mineev-Weinstein, Mark

PY - 2017/7/1

Y1 - 2017/7/1

N2 - 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.

AB - 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.

KW - Diffusion-limited aggregation

KW - Laplacian growth

KW - Non-equilibrium growth processes

KW - Statistical physics

UR - http://www.scopus.com/inward/record.url?scp=85018670109&partnerID=8YFLogxK

U2 - 10.1007/s10955-017-1796-9

DO - 10.1007/s10955-017-1796-9

M3 - Article

AN - SCOPUS:85018670109

VL - 168

SP - 68

EP - 91

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -