Abstract
The aim of this year's lecture is to describe random hyperbolic surfaces, their geometry and spectrum.
We will also discuss random regular graphs, whose combinatorics and spectral theory are in many ways analogous to those of hyperbolic surfaces. In this session, we introduce the probabilistic method of Paul Erdös, and define several models of random regular graphs. We're mainly interested in their Cheeger constant and spectral hole, but also in their systole and diameter.
References
P. Erdös, Graph theory and probability. Canadian Journal of Math. 1959. Proof of inequality (4)
B. Bollobás, The isoperimetric number of random regular graphs, Eur. Journal of Combinatorics 1988
P. Diaconis, D. Stroock, Geometric bounds for eigenvalues or Markov chains, Ann. Appl. Probab. 1991. Proposition 6 (Cheeger inequality).
J. Friedman, A Proof of Alon's Second Eigenvalue Conjecture and Related Problems. Memoirs of the AMS 2008. Introduction (Part 1, definition of regular random graph models)
