Nicholas Georgiou

Deposition, diffusion, and nucleation on an interval

We study a model motivated by nanoscale growth of ultra-thin films. Particles are deposited on an interval substrate (at rate $\lambda > 0$) where they perform Brownian motions until any two meet, at which point they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. We focus on the interval-splitting process induced by the sequence of nucleations. We show that:

(i) as $\lambda \to 0$ the process converges to a Markovian interval-splitting process, governed by a splitting density which we describe;

(ii) the same splitting density also governs the fixed-$\lambda$, large-time asymptotics of the normalised gap distribution;

(iii) this splitting density can be derived from the solution to an exit problem for planar Brownian motion from a right-angled triangle.

Based on joint work with Andrew Wade.