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Abstract
In this paper we consider the expansion properties and the spectrum of the combinatorial Laplace operator of a ddimensional LinialMeshulam random simplicial complex, above the cohomological connectivity threshold. We consider the spectral gap of the Laplace operator and the Cheeger constant as this was introduced by Parzanchevski, Rosenthal and Tessler (Combinatorica 36, 2016). We show that with high probability the spectral gap of the random simplicial complex as well as the Cheeger constant are both concentrated around the minimum codegree of among all (d − 1)faces. Furthermore, we consider a random walk on such a complex, which generalises the standard random walk on a graph. We show that the associated conductance is with high probability bounded away from 0, resulting in a bound on the mixing time that is logarithmic in the number of vertices of the complex.
Original language  English 

Journal  Random Structures and Algorithms 
Early online date  16 Jul 2021 
DOIs  
Publication status  Epub ahead of print  16 Jul 2021 
Keywords
 Cheeger constant
 Laplace operator
 conductance
 random simplicial complexes
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Dive into the research topics of 'Algebraic and combinatorial expansion in random simplicial complexes'. Together they form a unique fingerprint.Projects
 1 Finished

Dynamic models of random simplicial complexes
Engineering & Physical Science Research Council
3/12/17 → 2/12/20
Project: Research Councils