One-arm domination time in Cylindrical Hastings-Levitov$(0)$

TL;DR

This paper analyzes the growth dynamics of a single-arm structure in the cylindrical Hastings-Levitov(0) model, providing bounds on the expected time for the arm to receive a particle and demonstrating exponential tail decay.

Contribution

It establishes precise bounds on the expected last particle reception time and proves exponential tail decay for the cylindrical Hastings-Levitov(0) model's arm growth.

Findings

Expected last particle reception time scales as N^2 / λ^3

Expected time bounds are proportional to N^2 / λ^3

Exponential tail decay for the last reception time

Abstract

The cylindrical Hastings-Levitov$(0)$ admits a single infinite connected tree (arm). For a cylinder of width $N$, and particles of size $\lambda$, we consider the last time $\upsilon_{N, \lambda}$, that a finite tree receives a particle. We prove that $\frac{cN^2}{\lambda^3}<\mathbb{E}[\upsilon_{N, \lambda}]< \frac{CN^2}{\lambda^3}$. Furthermore, we establish an exponential tail for $\upsilon_{N, \lambda}$.