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

TL;DR

The paper analyzes HL(0) growth in the small-particle limit by embedding the dynamics in a Loewner–Kufarev framework and proving a large deviation principle for the driving measures with rate given by relative entropy. It characterizes which shapes can be generated by finite-entropy evolutions, tying entropy to geometry via Loewner energy and Weil–Petersson/Becker quasicircles, and demonstrates a rich set of finite-entropy examples including cusps and non-simple curves. It then explores optimization questions: under a shape constraint, what driving measure minimizes entropy, and how this relates to (and sometimes diverges from) entropy-minimizing Loewner-energy foliations; it also presents a transport-equation–type simplification for non-analytic Loewner dynamics. The results connect probabilistic large deviations, complex-analytic growth, and geometric function theory, providing a framework to quantify how rare HL(0) growth patterns deviate from concentric discs and which shapes remain attainable with finite entropy.

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 $ρ$ for the Loewner--Kufarev equation: \[ H(ρ) = \frac{1}{2π}\iint \barρ_t(θ) \log \barρ_t(θ) dθdt, \] whenever $ρ= \barρ_t dθdt/2π$ with $\int_{S^1} \barρ_t dθ/2π= 1$. 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.

Figures (6)

• Figure 1: The interval $J_{i}^{n,m}= \left[ \left\lceil\frac{i-1}{m}n\right\rceil\frac{1}{n}, \left\lfloor\frac{i}{m}n\right\rfloor\frac{1}{n}\right)$ consists of those intervals of length $1/n$ that are fully contained in $[\frac{i-1}{m},\frac{i}{m})$.
• Figure 2: The growing hull given by the map $f_t: \Delta \to \Delta\smallsetminus B(e^t, \sqrt{e^{2t}-1})$
• Figure 3: The hulls grown by the conformal map $\tilde{f}_{s}:\Delta \to \Delta\smallsetminus B(\frac{1}{1-s}, \frac{1+s}{1-s})$. Note that the interfaces are not Jordan curves.
• Figure 4: The hulls generated by the Poisson kernel density with constant parameter $R>1$.
• Figure 5: The hulls grown by the Poisson kernel density with different choices of $R=R(t)$.

Theorems & Definitions (45)

• Remark
• Theorem 1.1
• Remark
• Remark
• Proposition 1.2
• Corollary 1.3
• proof
• Remark
• Definition 1.5
• Definition 1.6