On the Growth of a Ballistic Deposition Model on Finite Graphs

TL;DR

This paper investigates a ballistic deposition process on finite graphs, analyzing its growth rate, fluctuations, and providing new graph-theoretic insights, with results including growth parameters and a central limit theorem.

Contribution

It determines the asymptotic growth parameter for certain graphs, proves a central limit theorem for fluctuations, and offers a novel graph-theoretic interpretation of existing inequalities.

Findings

Determined the growth parameter b3(6b1) for some graphs

Proved a central limit theorem for the process fluctuations

Provided a new graph-theoretic interpretation of an existing inequality

Abstract

We revisit a ballistic deposition process introduced by Atar, Athreya and Kang. Let $\mathcal{G}=(V,E)$ be a finite connected graph. We choose independently and uniformly vertices in $\mathcal{G}$. If a vertex $x$ is chosen and the previous height configuration is given by $h=(h_y)_{y \in V} \in \mathbb{N}_0^V$, the height $h_x$ is replaced by \[ \tilde{h}_x := 1 + \max_{y \sim x} h_y. \] We study asymptotic properties of this growth model. We determine the asymptotic growth parameter $\gamma(\mathcal{G} )$ for some graphs and prove a central limit theorem for the fluctuations around $\gamma ( \mathcal{G})$. We also give a new graph-theoretic interpretation of an inequality obtained by Atar et al..