IS28: Random Matrices and Combinatorial Structures

Organizer: Elia Bisi (University of Florence)

The spectrum of dense kernel-based random graphs

Michele Salvi

We study a broad class of inhomogeneous spatial random graphs, including long-range and scale-free percolation and preferential attachment-like models (see session IS07 for more details!). Vertices are placed on the discrete d-dimensional torus and are equipped with heavy tailed random weights. The probability of linking any pair of vertices decays in their distance but increases as a function of the weights. We focus on the adjacency matrix of such graphs in the dense regime and prove that, as the size of the torus goes to infinity, the empirical spectral distribution converges. The corresponding limiting measure is given by an operator-valued semicircle law that we show to be absolutely continuous and to have finite second moment, even when the weights have infinite variance. We characterize its Stieltjes transform by a fixed point equation in an appropriate Banach space.

Based on a joint work with A. Cipriani, R. S. Hazra and N. Malhotra.