Logarithmic fluctuations of Stationary Hastings-Levitov

TL;DR

This paper demonstrates that the fluctuation field of stationary Hastings-Levitov$(0)$ has logarithmic spatial correlations and establishes bounds on the growth of the imaginary part of the field over time.

Contribution

It provides a rigorous analysis of the logarithmic correlations and growth bounds in the fluctuation field of stationary Hastings-Levitov$(0)$, a model in complex analysis.

Findings

Fluctuation field exhibits logarithmic spatial correlations.

The imaginary part grows at most logarithmically in time with high probability.

Provides new insights into the behavior of stationary Hastings-Levitov models.

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 $\beta>0$, $\max_{x\in[0,t]}\text{Im} M_t(x)<\beta\log t$ with high probability, as $t\to\infty$.