MARK KAC SEMINAR

The first part is an introduction to arctic phenomena in random models, namely the emergence of deterministic boundaries, called arctic curves, between regions which show very different qualitative behaviours. Historically, the first incarnation of this phenomenon has been found in the now famous Aztec diamonds, for which the arctic circle theorem was rigourously established in 1998. Many other models with similar features have been subsequently studied. The second part focuses on the so-called Tangent Method, an efficient method to compute the shape of arctic curves. The principles and main assumptions underlying the method will be presented, as well as its limitations. The method will then be illustrated in the specific case of Aztec diamonds.

In the first part of the talk, I will describe some results on the bulk of the spectrum of sparse and dense inhomogeneous random graphs and how the two spectrums are related. In the second part of the talk, I will focus on the edge on the spectrum and describe properties of the largest eigenvalue, eigenvectors and some related large deviation results.