Quantized Laplacian growth, II: 1D hydrodynamics of the Loewner density

TL;DR

This paper provides an analytic framework for understanding fluctuations in Laplacian growth, showing how microscopic regularization leads to chaotic interface dynamics and universal pattern formation.

Contribution

It introduces a regularization mechanism that models microscopic fluctuations and derives their evolution using Dyson Brownian motion and the viscous Burgers equation.

Findings

Fluctuations cause chaotic interface behavior.

Universal patterns like fjords and fingers emerge asymptotically.

Microscopic regularization prevents cusp formation in finite time.

Abstract

A systematic analytic treatment of fluctuations in Laplacian growth is given. The growth process is regularized by a short-distance cutoff $\hbar$ preventing the cusps production in a finite time. This regularization mechanism generates tiny inevitable fluctuations on a microscale, so that the interface dynamics becomes chaotic. The time evolution of fluctuations can be described by the universal Dyson Brownian motion, which reduces to the complex viscous Burgers equation in the hydrodynamic approximation. Because of the intrinsic instability of the interface dynamics, tiny fluctuations of the interface on a microscale generate universal patterns with well developed fjords and fingers in a long time asymptotic.