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Random matrices and permutations, matrix integrals and integrable systems. (English) Zbl 0995.15019
Séminaire Bourbaki. Volume 1999/2000. Exposés 865-879. Paris: Société Mathématique de France, Astérisque 276, 411-433, Exp. No. 879 (2002).
This lecture gives a survey of recent interactions between the theory of random matrices, random permutations, and integrable models. Two sets of integrals form the basis of the lecture. It contains the following chapters: (a) Largest increasing sequences in random permutations; (b) The spectrum of random matrices; (c) Infinite Hermitian matrix ensembles; (d) Large random matrices and permutations: a direct connection via enumerative geometry; (e) Integrals, moment matrices and integrable systems.
For the entire collection see [Zbl 0981.00011].

MSC:
15B52 Random matrices (algebraic aspects)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
05A05 Permutations, words, matrices
14H70 Relationships between algebraic curves and integrable systems
37K60 Lattice dynamics; integrable lattice equations
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
60C05 Combinatorial probability
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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