Spectral large deviations of sparse random matrices

par Ella Hiesmayr

jeudi 25 sept. 2025, 14:00 → 15:00 Europe/Paris

435 (ENS)

In this talk we consider the adjacency matrix of Erdős-Rényi graphs with constant average degree, that are equipped with i.i.d. edge weights. We determine the large deviations of the largest eigenvalue for those graphs, with particular interest in the effect of the weight distribution. To this end, we consider random variables that have lighter, and heavier tails than the Gaussian distribution. Surprisingly in the light-tailed case the rate function is universal, whereas in the heavy-tailed case it depends on the precise entry distribution, and is given in the form of a variational problem. In our proofs we rely on a precise analysis of the geometry of the graph, as more general methods break down in this regime of sparsity.