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Date
2015-09-07Type
- Working Paper
ETH Bibliography
yes
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Abstract
We consider the adjacency operator of the Linial-Meshulam model for random simplicial complexes on n vertices, where each d-cell is added independently with probability p to the complete (d−1)-skeleton. Under the assumption np(1−p)≫log4n, we prove that the spectral gap between the (n−1d) smallest eigenvalues and the remaining (n−1d−1) eigenvalues is np−2√dnp(1−p)(1+o(1)) with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. The main ingredient of the proof is a Füredi-Komlós-type argument for random simplicial complexes, which may be regarded as sparse random matrix models with dependent entries. Show more
Publication status
publishedExternal links
Journal / series
arXivPages / Article No.
Publisher
Cornell UniversityOrganisational unit
09456 - Knowles, Antti (SNF-Professur) (ehemalig)
03900 - Nolin, Pierre
Funding
144662 - Spectral and eigenvector statistics of large random matrices (SNF)
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ETH Bibliography
yes
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