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Deposition, diffusion, and nucleation on an interval

Georgiou, Nicholas; Wade, Andrew R.

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Abstract

Motivated by nanoscale growth of ultra-thin films, we study a model of deposition, on an interval substrate, of particles that perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model studied in the applied literature. We show that the associated interval-splitting process converges in the sparse deposition limit to a Markovian process (in the vein of Brennan and Durrett) governed by a splitting density with a compact Fourier series expansion but, apparently, no simple closed form. We show that the same splitting density governs the fixed deposition rate, large time asymptotics of the normalized gap distribution, so these asymptotics are independent of deposition rate. The splitting density is derived by solving an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson.

Citation

Georgiou, N., & Wade, A. R. (2022). Deposition, diffusion, and nucleation on an interval. Annals of Applied Probability, 32(6), 4849-4892. https://doi.org/10.1214/22-aap1804

Journal Article Type Article
Acceptance Date Feb 27, 2022
Online Publication Date Dec 6, 2022
Publication Date 2022-12
Deposit Date Aug 9, 2021
Publicly Available Date Dec 16, 2022
Journal Annals of Applied Probability
Print ISSN 1050-5164
Publisher Institute of Mathematical Statistics
Peer Reviewed Peer Reviewed
Volume 32
Issue 6
Pages 4849-4892
DOI https://doi.org/10.1214/22-aap1804

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