Integrable Lattices: Random Matrices and Random Permutations
Abstract
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 timeparameters, are natural taufunctions 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2000
 arXiv:
 arXiv:math/0010135
 Bibcode:
 2000math.....10135V
 Keywords:

 Mathematics  Combinatorics;
 Mathematical Physics;
 Mathematics  Mathematical Physics;
 Mathematics  Probability;
 Condensed Matter;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 "Random Matrices and Their Applications" : MSRIpublication #40, Cambridge University Press, 2001