Abstract
A notorious problem in mathematics and physics is to create a solvable model for random sequential adsorption of non-overlapping congruent spheres in the d-dimensional Euclidean space with dge 2. Spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres. Due to spatial correlations, characterizing the fraction of accepted spheres remains largely intractable. We study this fraction by taking a novel approach that compares random sequential adsorption in Euclidean space to the nearest-neighbor blocking on a sequence of clustered random graphs. This random network model can be thought of as a corrected mean-field model for the interaction graph between the attempted spheres. Using functional limit theorems, we characterize the fraction of accepted spheres and its fluctuations.
Highlights
Random sequential adsorption of congruent spheres in the d-dimensional Euclidean space has been a topic of great interest across the sciences, serving as basic models in condensed matter and quantum physics [21,28,35,37], nanotechnology [11,14], information theory and optimization problems [20,24,40]
We introduced a clustered random graph model with tunable local clustering and a sparse superimposed structure
The level of clustering was set to suitably match the local clustering in the topology generated by the random geometric graph
Summary
Random sequential adsorption of congruent spheres in the d-dimensional Euclidean space has been a topic of great interest across the sciences, serving as basic models in condensed matter and quantum physics [21,28,35,37], nanotechnology [11,14], information theory and optimization problems [20,24,40]. The fraction of accepted spheres can be obtained via the following greedy algorithm to find independent sets of rgg: Given a graph G, initially, all the vertices are declared inactive. We introduce an approximate approach for studying J(c, d) and V(c, d) by considering rsa on a clustered random graph model, designed to match the local spatial properties of the rgg model in terms of average degree and clustering. Exact analysis of this random graph model leads to expressions for the limiting jamming fraction and its fluctuations, in turn providing approximations for J(c, d) and V(c, d).
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