Integral Operators in Random Matrix Theory

Harold Widom

Abstract:

Lecture 1. (Monday, July 4) Hilbert-Schmidt and trace class operators. Definitions and fundamental properties of the trace and determinant. Fredholm determinants of integral operators.

Lecture 2. (Wednesday, July 6) How integral operators arise in random matrix theory. Correlation functions and kernels of integral operators. Spacing distributions as operator determinants. Scaling of the Gaussian ensembles. The sine and Airy kernels.

Lecture 3. (Friday, July 8) Use of operator determinants to derive differential equations for distribution functions arising in random matrix theory. The representations of basic distributions in terms of Painlevé functions.