Abstract
This paper concerns the long term behaviour of a growth model describing a random sequential allocation of particles on a finite cycle graph. The model can be regarded as a reinforced urn model with graph-based interaction. It is motivated by cooperative sequential adsorption, where adsorption rates at a site depend on the configuration of existing particles in the neighbourhood of that site. Our main result is that, with probability one, the growth process will eventually localise either at a single site, or at a pair of neighbouring sites.
Highlights
This paper concerns a probabilistic model describing a sequential allocation of particles on a finite cycle graph
The model is motivated by cooperative sequential adsorption (CSA)
CSA models are widely applied in physical chemistry
Summary
This paper concerns a probabilistic model describing a sequential allocation of particles on a finite cycle graph. In the case of no interaction, in which the growth rate at site i is given by Γ (xi ), where xi is the number of existing particles at site i and Γ : Z+ → (0, ∞) is a given function (called the reinforcement rule [4] or feedback function [12]), our model coincides with a generalised Pólya urn (GPU) model with a particular reinforcement rule Γ. In our growth model the rate of growth at site i is given by a site-dependent reinforcement rule Γi = exp(λi ui ), where λi > 0 and ui is the number of existing particles in a neighbourhood of site i This allows one to take into account the case where different sites might possibly have different reinforcement schemes (Fig. 2). These results combined with stochastic domination techniques are constantly used in the proofs of Lemmas 5–8
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