Document Type
Lecture
Publication Date
10-10-2014
Abstract
Random matrix theory (RMT) is a very active area of research and a great source of exciting and challenging problems for specialists in many branches of analysis, spectral theory, probability, and mathematical physics. The analysis of the eigenvalue distribution of many random matrix ensembles leads naturally to determinantal point processes and, in particular, to biorthogonal ensembles where the correlation kernel is written in terms of two sequences of mutually orthogonal functions. Another source of determinantal point processes is a class of stochastic models of particles following non-intersecting paths. The connection of these models with RMT is tight: for example, the eigenvalues of the Gaussian Unitary Ensemble (GUE) and the distribution of Brownian particles conditioned not to intersect are statistically identical. A great challenge is the detailed asymptotic analysis of these processes as the number of particles or matrix size grows. One powerful tool is the Riemann-Hilbert characterization of the correlation kernel and the associated non-commutative steepest descent analysis of Deift and Zhou. Some ideas behind this technique are illustrated using squared Bessel nonintersecting paths.
Relational Format
presentation
Recommended Citation
Martinez-Finkelshtein, Andrei, "Random Matrix Models, Non-intersecting random paths, and the Riemann-Hilbert Analysis" (2014). Colloquium. 22.
https://egrove.olemiss.edu/math_colloquium/22
