Some results on beta ensembles and on scaling limits of random graphs

Abstract

This thesis is divided into two main parts, each of which can be studied independently of the other. The first part is devoted to the study of beta ensembles (β-ensembles). The main theme in β-ensembles, in this thesis, is the study of the largest eigenvalues. In particular we study their tail behaviour, stochastic domination and log-concavity properties. The second part deals with the scaling limits of critical random graphs. We obtain the scaling limit of sizes and intrinsic geometry of the connected components of critical random graph models, converging in a suitable sense to an L3 graphon. β-ensembles: These random matrix models, starting from the work of Dyson, have connections to several models such as last passage percolation, random polymer growth, random permutations. The largest eigenvalues of β-ensembles are related to the Tracy-Widom β (TWβ) distributions. Ram´ırez-Rider-Vir´ag showed that the fluctuations of the largest eigenvalues of Hermite and Laguerre β-ensembles converge to TWβ. We establish tail estimates for the largest eigenvalues of Hermite and Laguerre β- ensembles optimal up to the constants in the exponents. These estimates strengthen the results of Ledoux-Rider and have several applications. We also prove that the largest eigenvalues exhibit stochastic domination which is a β generalization of the stochastic domination known for last passage times. We also study log-concavity in β-ensembles (both continuous and discrete). Our results allow us to prove several properties of last passage times, TWβ and prove Poissonized version of a conjecture of Chen on longest increasing sub-sequences. Scaling limits of random graphs: Aldous showed in a seminal work that inside the critical window, the rescaled sequence of component sizes of Erd˝os-R´enyi random graphs converge to a limit. Addario-Berry, Broutin and Goldschmidt showed that the rescaled components, viewed as metric spaces, converge to a sequence of random fractals. This scaling limit is expected to be universal for a large class of critical random graphs exhibiting mean field behaviour. We prove that under some regularity conditions, the critical percolation scaling limit of random graphs that converge to an L3 graphon in a suitable sense, is same as that of the Erd˝os-R´enyi random graphs. This is done by applying a general universality technique, due to Bhamidi, Broutin, Sen and Wang, for proving scaling limits of critical random graphs.