Abstract

Scale free percolation on a lattice is an important random graph model which considers inhomogeneity as well as spatial location of the vertices. We consider the model on a torus in arbitrary dimensions. The model assigns iid random weights on the vertices and conditional on these weights, we connect any two points on the torus with probability which depends on the weights and also the torus distance between the vertices. The percolation properties have been analysed in recent years and phase transitions occurs depending on the certain parameters. We focus on the adjacency operator of this random graph and study some interesting properties of the spectrum. We identify the spectrum explicitly in some cases. This is based on a joint work with A.Cipriani (University College London), Nandan Malhotra (Leiden University) and Michele Salvi (Tor Vergata, Rome).