Loewner--Kufarev entropy and large deviations of the Hastings--Levitov model

TL;DR

This paper establishes a large deviation principle for the Hastings--Levitov HL(0) model in the small particle limit, linking the rate function to the relative entropy of the driving measure, and explores the class of shapes generated by finite entropy Loewner evolution.

Contribution

It introduces a large deviation principle for the HL(0) model and characterizes the shapes generated by finite entropy Loewner evolution, including various classes of quasicircles and curves.

Findings

Large deviation principle with relative entropy rate function

Shapes include all Weil-Petersson and Becker quasicircles, plus more complex curves

Analysis of minimal entropy measures for shape generation

Abstract

We consider the Hastings--Levitov HL(0) model in the small particle scaling limit and prove a large deviation principle. The rate function is given by the relative entropy of the driving measure $\rho$ for the Loewner--Kufarev equation: \[ H(\rho) = \frac{1}{2\pi}\iint \bar{\rho}_t(\theta) \log \bar{\rho}_t(\theta) d\theta dt, \] whenever $\rho = \bar{\rho}_t d\theta dt/2\pi$ with $\int_{S^1} \bar{\rho}_t d\theta/2\pi = 1$. We investigate the class of shapes that can be generated by finite entropy Loewner evolution and show that it contains all Weil-Petersson quasicircles, all Becker quasicircles, a Jordan curve with a cusp, and a non-simple curve. We also consider the problem of finding a measure of minimal entropy generating a given shape as well as a simplified version of the problem for a related transport equation.