Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials. {T}he continuous case (Q1767521)

Empirical spectral distributions of complex random matrices from the Gaussian, Laguerre, and Jacobi unitary ensembles are studied by means of methods, based on the research by \textit{U. Haagerup} and \textit{S. Thorbjørnsen} [Expo. Math. 21, No. 4, 293--337 (2003; Zbl 1041.15018)]. In the framework of abstract Markov operators [see \textit{D. Bakry}, in: Lectures on probability theory. Lect. Notes Math. 1581, 1--114 (1994; Zbl 0856.47026)], differential equations for Laplace transforms of eigenfunction measures are obtained, which allows one to get the recurrence relations for the moments of these measures. The latter ones are employed to describe sharp small deviation inequalities on the largest eigenvalues.