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.
Acknowledgments
The authors are grateful to Michael Grinfeld for introducing them to deposition and nucleation models, and to Michael Grinfeld and Paul Mulheran for stimulating discussions on this topic over several years. The authors also thank an anonymous referee for a careful reading and helpful remarks.
Citation
Nicholas Georgiou. Andrew R. Wade. "Deposition, diffusion, and nucleation on an interval." Ann. Appl. Probab. 32 (6) 4849 - 4892, December 2022. https://doi.org/10.1214/22-AAP1804
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