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 concerning the scaling limits of random graphs. In a seminal work, Aldous (Ann. Probab., 1997) showed that inside the critical window, the rescaled sequence of component sizes of Erdős–Rényi random graphs converge to a limit. This limiting sequence is the sequence of excursion lengths of a reflected inhomogeneous Brownian motion with a negative drift. Addario-Berry, Broutin and Goldschmidt (Probab. Theory Related Fields, 2012) showed that the rescaled components, viewed as metric spaces, converge to a sequence of random fractals. We prove that under some regularity conditions, the critical percolation scaling limit of random graphs that converge to an L^3 graphon in a suitable sense, is same as that of the Erdős–Rényi random graphs.

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.