M.Sc Student M.Sc Thesis Leibzirer Shaked On the Eigenvalues in the Linial-Meshulam Model Department of Mathematics ASSOCIATE PROF. Ron Rosenthal

Abstract

Denote by the adjacency operator of , the Linial-Meshulam model for random dimensional simplicial complexes on vertices, where each cell is added independently with probability  to the complete skeleton. We consider the matrix , a sparse random matrix, which generalizes , the centered and normalized adjacency matrix of . While the non-zero entries of are normalized Bernoulli random variables, those of  are any bounded random variables with the same law. We show that for any positive constant , all large enough (depending on  and ) and any integer satisfies  , we have that , where denotes the Schatten norm, are explicit functions of the entries of , and  is an explicit function of  and . We use this bound in order to prove that under the assumption , one has , and   almost surely, where  is a random variable with the same law as the entries of . The main tool of the proof is a generalization of a result by Latala, Van Handel and Youssef from 2018, which is based on combinatorial arguments and the exploitation of the dependent structure of the entries of , which is governed by the simplicial structure.