Random geometric complexes in the thermodynamic regime

By D. Yogeshwaran, Eliran Subag, Robert J. Adler. Probability theory and related fields, 2014

96 citations recorded by the source corpus.

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called ‘thermodynamic’ regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer–Vietoris exact sequences. The Mayer–Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/2e42774b3ca0d2b6d71a3763471f33d0c5094f4d

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In Applied Mathematics

In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

On the topology of random complexes built over stationary point processes. (2012, methodology): “The above asymptotics have been strengthened to convergence and strong laws in the recent preprint [41].”
Maximally Persistent Cycles in Random Geometric Complexes (2015, background): “Several recent papers have studied various aspects of the topology of these complexes (see [6, 9, 11, 38, 40, 50, 51] and the survey [8]).”