Random Balanced Cayley Complexes
Roy Meshulam

TL;DR
This paper extends the Alon-Roichman theorem to higher-dimensional complexes, showing that random subsets of a finite group produce complexes with spectral gaps, indicating high-dimensional expansion properties with high probability.
Contribution
It proves a k-dimensional analogue of the Alon-Roichman theorem, establishing spectral gap bounds for random Cayley complexes.
Findings
Spectral gap bounds hold with high probability for random subsets.
The size of the subset depends logarithmically on the sum of degrees of irreducible representations.
The result generalizes known expansion properties to higher-dimensional simplicial complexes.
Abstract
Let be a finite group of order and for let . Viewing each as a -dimensional complex, let denote the simplicial join . For let be the subcomplex of that contains the -skeleton of and whose -simplices are all such that . Let denote the reduced -th Laplacian of , acting on the space of real valued -cochains of . The -th spectral gap of is the minimal eigenvalue of . The following -dimensional analogue of the Alon-Roichman theorem is proved: Let and be fixed and let be a random subset of of size $m= \left\lceil\frac{10 k^2\log…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Limits and Structures in Graph Theory · Quasicrystal Structures and Properties
