Integrable Lattices: Random Matrices and Random Permutations
van Moerbeke, Pierre
arXiv, 0010135 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2000-10-13
Classification:  Mathematics - Combinatorics,  Condensed Matter,  Mathematical Physics,  Mathematics - Probability,  Nonlinear Sciences - Exactly Solvable and Integrable Systems 
@article{0010135,
     author = {van Moerbeke, Pierre},
     title = {Integrable Lattices: Random Matrices and Random Permutations},
     journal = {arXiv},
     volume = {2000},
     number = {0},
     year = {2000},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0010135}
}

van Moerbeke, Pierre. Integrable Lattices: Random Matrices and Random Permutations. arXiv, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/0010135/