Some results about ergodicity in shape for a crystal growth model

TL;DR

This paper investigates the conditions under which a crystal growth Markov model exhibits ergodicity or transience, providing improved criteria and detailed asymptotic behavior in a specific case.

Contribution

It improves existing conditions for ergodicity and transience in a crystal growth model and offers a detailed analysis of asymptotic behavior in a particular scenario.

Findings

Established improved conditions for ergodicity.

Identified conditions leading to transience.

Described asymptotic behavior in a special case.

Abstract

We study a crystal growth Markov model proposed by Gates and Westcott (\cite{Kinetics1}, \cite{Kinetics2}). This is an aggregation process where particles are packed in a square lattice accordingly to prescribed deposition rates. This model is parametrized by three values $(\beta_i, i=0,1,2)$ corresponding to depositions on three different types of sites. The main problem is to determine, for the shape of the crystal, when recurrence and when ergodicity do occur. In \cite{AMS} and \cite{MarkovModels} sufficient conditions are given both for ergodicity and transience. We establish some improved conditions and give a precise description of the asymptotic behavior in a special case.