Integrable Lattices : Random Matrices and Random Permutations ∗ Pierre

These lectures present a survey of recent developments in the area of random matrices (finite and infinite) and random permutations. These probabilistic problems suggest matrix integrals (or Fredholm determinants), which arise very naturally as integrals over the tangent space to symmetric spaces, as integrals over groups and finally as integrals over symmetric spaces. An important part of these lectures is devoted to showing that these matrix integrals, upon apropriately adding time-parameters, are natural tau-functions for integrable lattices, like the Toda, Pfaff and Toeplitz lattices, but also for integrable PDE’s, like the KdV equation. These matrix integrals or Fredholm determinants also satisfy Virasoro constraints, which combined with the integrable equations lead to (partial) differential equations for the original probabilities. Expanded version of lectures at MSRI, Berkeley, Spring 1998. To appear in ”Random Matrices and Their Applications” : Mathematical Sciences Research Institute Publications #40, Cambridge University Press, 2001. Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium and Brandeis University, Waltham, Mass 02454, USA. E-mail: [email protected] and @math.brandeis.edu. The support of a National Science Foundation grant # DMS-98-4-50790, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged. 1 §0, p.2