Matrix Integrals,Combinatorics and Integral Lattices

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Abstract for DMS - 0100782(1) What is the distribution of the eigenvalues of arandom matrix, with certain symmetry conditions toguarantee the reality of the spectrum? What happens tothe distribution, when the size of the matrix getslarge? What about universality in the limit?(2) What is the statistics of the length of thelongest increasing sequence in random permutation orrandom words (Ulam's problem). This question appliesto models of interface growth, polymers in randomenvironments, first passage percolation problem, and "dimer" configurations.(3) Integrals over groups and symmetric spaces, (orover their tangent spaces) lead to a variety ofinteresting matrix models. The coefficients of the(perturbative) expansions have striking combinatorialor topological significance and can be computed in arecursive way. This work originates in the works ofFeynman, 't Hooft, Bessis-Itzykson-Zuber and Witten,in the context of string theory.(4) The sample canonical correlation coefficients(maximum likelihood estimates) for the canonicalcorrelation coefficients of two Gaussian populationsare the test statistic for the statisticalindependence of the two populations, as studied byHotelling, James and Constantine.(5) The four problems above and their"time"-perturbations are all solutions to integrableequations or lattices. For large random matrices orpermutations, they are solutions to the Korteweg-deVries equation. In the finite case, they are solutionsto the Toda lattice, and to two new integrablelattices, 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 wayto new integrable systems, but also to newcombinatorial and probabilistic questions !General description: The problems aboverelate to a number of important questions in physics,engineering and statistics:Problem (1) has its origin in the study of energylevels (excitation spectra) of heavy nuclei in nuclearphysics (Wigner, Dyson). These levels are so intricatethat any explicit description would be intractable.For that reason, Wigner proposed a statistical modelfor 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 tostatistical 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, mentionedabove, are used in testing whether two sets ofstatistical data are correlated, as sketched in (4).The Korteweg-de Vries equation, the arch-type``soliton equation", describes the propagation ofwaves in shallow water; a similar non-linear partialdifferential equation governs the propagation ofultra-short pulses in optical fibers, when the wavelength is long compared to the diameter of the fiber.