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Extreme Eigenvalues of critical Erdoes-Rényi graphs

Date: Thursday, July 11, 2019 15:30 - 18:00
Speaker: Johannes Alt (University of Geneve)
Location: Heinzel Seminar Room / Office Bldg West (I21.EG.101)
Series: Mathematics and CS Seminar
Host: Laszlo Erdös

Abstract:

In this talk, we present recent results on the extreme eigenvalues of the adjacency matrix of Erd?s-Rnyi graphs. The Erd?s-Rnyi graph G has N vertices and any two vertices are connected with probability p, independently of other edges.
If p is large then the adjacency matrix A of G behaves like a Wigner random matrix and has the semicircle law on [-2,2] as limiting eigenvalue density. Moreover, the extreme eigenvalues converge to -2 and 2, respectively.
If p is small then, however, A has many eigenvalues outside [-2,2].
Recently, the critical value of p for this transition has been determined and a precise connection between the large degrees of G and the extreme eigenvalues of A has been established.
This is joint work with Raphael Ducatez and Antti Knowles.
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