Eigenvalue confinement and spectral gap for random simplicial complexes1. Issue 3 (24th April 2017)

• Record Type: Journal Article
• Authors: Knowles, Antti; Rosenthal, Ron
• Is Part Of: Random structures & algorithms. Volume 51:Issue 3(2017)
• Page Start: 506
• Page End: 537
• Publication Date: 2017-04-24
• Subjects: #random simplicial complexes -- Linial Meshulam model -- random matrices -- spectral gap; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20710
• Languages: English
• ISSNs: 1042-9832

Abstract

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 n p ( 1 − p ) ≫ log 4 n, we prove that the spectral gap between the ( n − 1 d ) smallest eigenvalues and the remaining ( n − 1 d − 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