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This talk is broadly divided into three parts. We first discuss on hole probabilities of the finite and infinite Ginibre ensembles, Beta ensembles and the Mittag-Leffler determinantal point processes in the complex plane. The hole probability means the probability that there is no points in a given region for a given point process. We study the asymptotic of the hole probabilities as the size of the region increases. The equilibrium measure plays a crucial role in calculating the hole probabilities. The equilibrium measure and the minimum energy related results will be discussed.
Second, we discuss on random matrix related results. We show that the eigenvalues of product of independent Ginibre matrices form a determinantal point process in the complex plane. A matrix with iid standard complex normal entries is known as Ginibre matrix. Then we introduce basic notion of free probability and Brown measure. We show that the limiting spectral distribution of the product of elliptic matrices is same as the Brown measure of its limiting element (*-distribution sense). We calculate the limiting spectral distribution of the product of truncated unitary matrices using free probability and Brown measure techniques.
Finally, we discuss on the fluctuations of the linear statistics of the eigenvalues of circulant, reverse circulant, symmetric circulant, Hankel, and variance profile random matrices. We re-establish some existing results on fluctuations of linear statistics of the eigenvalues by choosing appropriate variance profiles. |