In this talk we consider the adjacency matrix of Erdos-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.
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