Title: Spectrum of Linial-Meshulam complex in thermodynamic regime

 

Linial-Meshulam complex, denoted by , is a random simplicial complex on  vertices with a complete  -dimensional skeleton and -simplices chosen independently with probability . Linial-Meshulam complex is one of the most studied generalizations of the Erdos-Renyi random graph in higher dimensions.

In this talk, we discuss the limiting spectral distribution of the adjacency matrices of the Linial-Meshulam complex when . We prove the existence of a non-random limiting spectral distribution and derive some properties of the limiting spectral distribution. We show that the spectrum is connected to the spectrum of the line graph of Linial-Meshulam complex. We also derive the local weak limit of the line graph and study its consequence on the continuous part of the spectrum. This talk is based on a joint work with Kartick Adhikari and Koushik Saha.