Logarithmic fluctuations of Stationary Hastings-Levitov

TL;DR

This work analyzes the stationary Hastings-Levitov (0) growth model in the half-plane and establishes that the fluctuation field M_t(x) has logarithmic spatial correlations. By decomposing the conformal slit maps and bounding the generator of the imaginary part, the authors obtain a precise variance growth E[|M_t(0)|^2] = (π/4) log t and a logarithmic covariance Cov(M_t(0), M_t(b)) = (π/4)(log t − log b). They further show maximal fluctuations in Im M_t over [0,t] are logarithmic in t with high probability. The results rely on exponential moment bounds from the SHL generator, Doob-type decompositions, and distortion bounds from conformal mapping theory, contributing to understanding logarithmic correlation structures in SHL(0).

Abstract

We prove that the fluctuation field $\{M_t(x)\}_{x\in\mathbb{R}}$ of stationary Hastings-Levitov$(0)$ exhibits logarithmic spatial correlations. Moreover, by studying the infinitesimal generator of the imaginary part of $M_t(0)$, we show that for some $β>0$, $\max_{x\in[0,t]}\text{Im} M_t(x)<β\log t$ with high probability, as $t\to\infty$.

Figures (4)

• Figure 1: Computer simulation of $\{\tilde{F}_t(x)\}_{x\in{\mathbb{R}}}$
• Figure 2: Computer simulation of $\{M_t(x)\}_{x\in{\mathbb{R}}}$
• Figure 3: $M_t(0)$ empirical distribution histogram
• Figure 4: Illustration of bounding set \ref{['eq:bounding set']}, with $z_l=\nu_{F_{2t}(-\frac{1}{2}t)}^{2t}(\tilde{t}_l)$ and $z_r=\nu_{F_{2t}(\frac{3}{2}t)}^{2t}(\tilde{t}_r)$.

Theorems & Definitions (30)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm:var']}
• Lemma 3.1
• proof
• Lemma 3.2
• proof