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Eigenvalue confinement and spectral gap for random simplicial complexes
Author(s) -
Knowles Antti,
Rosenthal Ron
Publication year - 2017
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20710
Subject(s) - mathematics , eigenvalues and eigenvectors , spectral gap , random matrix , adjacency matrix , combinatorics , simplicial approximation theorem , simplicial complex , operator (biology) , distribution (mathematics) , random graph , pure mathematics , discrete mathematics , mathematical analysis , quantum mechanics , physics , graph , simplicial set , biochemistry , chemistry , repressor , homotopy , transcription factor , homotopy category , gene
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 n p ( 1 − p ) ≫ log 4 n , we prove that the spectral gap between the(n − 1d)smallest eigenvalues and the remaining(n − 1d − 1)eigenvalues is n p − 2 d n p ( 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. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 506–537, 2017