ON THE GROWTH OF A BALLISTIC\\ DEPOSITION MODEL ON FINITE GRAPHS

#### G. Braun

2022, v.28, Issue 1, 1-28

ABSTRACT

We revisit a ballistic deposition process introduced by Atar et al.\ in \cite{AAK01}. Let $\mathcal{G}=(V,E)$ be a finite connected graph and choose independently and uniformly vertices in $\mathcal{G}$. If a vertex $x \in V$ gets chosen and the previous height configuration is $h=(h_y)_{y \in V} \in \mathbb{N}_0^V$, the height $h_x$ is replaced by

\[

\tilde{h}_x := 1 + \max_{y \sim x} h_y.

\]

For different underlying graphs $\mathcal{G}$, we determine the asymptotic growth parameter $\gamma(\mathcal{G} )$ of this model. We also present a central limit theorem for the height fluctuations around $\gamma ( \mathcal{G})$ and a graph-theoretic reinterpretation of an inequality obtained in \cite{AAK01}.

Keywords: ballistic deposition; random sequential adsorption; stochastic growth

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