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

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: \[ H(ρ) = \frac{1}{2π}\iint \barρ_t(θ) \log \barρ_t(θ) dθdt, \] 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.

31 pages, 5 figures