The thesis is divided into two main parts. The first part is devoted to the study of beta ensembles. In particular, we study the distribution of the rightmost point in beta ensembles, establishing new results on its tail behaviour, stochastic dominance, and log-concavity properties. The second part deals with the scaling limits of critical random graphs. We obtain the scaling limit of the sizes and intrinsic geometry of the connected components in critical random graph models, which generalize Erdős–Rényi random graphs.

For this talk, we focus solely on our results on beta 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. Ramírez-Rider-Virág 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.

The results on beta ensembles are based on joint works with Riddhipratim Basu, Sudeshna Bhattacharjee, Manjunath Krishnapur and Mokshay Madiman. The results on scaling limits of random graphs are based on a joint work with Shankar Bhamidi, Nicolas Broutin, Sanchayan Sen and Xuan Wang.