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
Seminar/Talk
Tags
Mathematics and CS Seminar, mathematical_seminar_ics
Host
Laszlo Erdös
Contact
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.