Random matrix theory and mathematical statistical mechanics

A Brownian bridge

Statistical physics and quantum physics are sources of deep and interesting stochastic models for complex systems. In some special cases these models can be solved exactly and this is often related to a rich mathematical structure, whereas in more general models one has to be satisfied with a less detailed understanding or numerical simulations. Many examples of models with an interesting mathematical structure come from random matrix theory and two-dimensional statistical mechanics.

Random matrices originated in nuclear physics in the 1950's as statistical models for complicated spectra. The eigenvalues of a random matrix, i.e. a matrix with random elements, provide models for typical behaviour of complex spectra. Since then random matrix theory has found applications in several areas of quantum physics. Another source of random matrices is the study of large data sets in statistics. Random matrix theory has interesting connections with two-dimensional statistical mechanical problems, e.g. random tiling/dimer models and random growing one-dimensional interfaces. There are connections to many other parts of mathematics for example enumerative combinatorics, integrable systems, number theory, spectral theory, special functions and probability theory.