Matrix Integrals,Combinatorics and Integral Lattices

Brandeis University, Waltham MA

Investigators

Abstract

Abstract for DMS - 0100782 (1) What is the distribution of the eigenvalues of a random matrix, with certain symmetry conditions to guarantee the reality of the spectrum? What happens to the distribution, when the size of the matrix gets large? What about universality in the limit? (2) What is the statistics of the length of the longest increasing sequence in random permutation or random words (Ulam's problem). This question applies to models of interface growth, polymers in random environments, first passage percolation problem, and "dimer" configurations. (3) Integrals over groups and symmetric spaces, (or over their tangent spaces) lead to a variety of interesting matrix models. The coefficients of the (perturbative) expansions have striking combinatorial or topological significance and can be computed in a recursive way. This work originates in the works of Feynman, 't Hooft, Bessis-Itzykson-Zuber and Witten, in the context of string theory. (4) The sample canonical correlation coefficients (maximum likelihood estimates) for the canonical correlation coefficients of two Gaussian populations are the test statistic for the statistical independence of the two populations, as studied by Hotelling, James and Constantine. (5) The four problems above and their "time"-perturbations are all solutions to integrable equations or lattices. For large random matrices or permutations, they are solutions to the Korteweg-de Vries equation. In the finite case, they are solutions to the Toda lattice, and to two new integrable lattices, the Pfaff and Toeplitz lattices. %Besides matrix integrals solutions, these lattices %have interesting rational solutions. It is fair to say that matrix integrals point the way to new integrable systems, but also to new combinatorial and probabilistic questions ! General description: The problems above relate to a number of important questions in physics, engineering and statistics: Problem (1) has its origin in the study of energy levels (excitation spectra) of heavy nuclei in nuclear physics (Wigner, Dyson). These levels are so intricate that any explicit description would be intractable. For that reason, Wigner proposed a statistical model for these energy levels. Concerning problem (2), it is well known that a large percentage of computer time is devoted to the rearrangement of the data used in the course of computations. How many data, after a complete reshuffling, are statistically still in order? This question of random permutations also applies to statistical mechanics, the basis of thermodynamics, and to questions of polymers. Some of the matrix models (3) provide toy models for string theory, an important set of ideas at the basis of the fundamental interactions in the physical world. It turns out that certain matrix models, mentioned above, are used in testing whether two sets of statistical data are correlated, as sketched in (4). The Korteweg-de Vries equation, the arch-type ``soliton equation", describes the propagation of waves in shallow water; a similar non-linear partial differential equation governs the propagation of ultra-short pulses in optical fibers, when the wave length is long compared to the diameter of the fiber.