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Near-extreme eigenvalues of random matrices and systems of coupled Painlevé II equations

Le : 23/01/2018 14h00
Par : Tom Claeys, Université Catholique de Louvain
Lieu : I 001
Lien web :
Résumé : For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy kernel determinantal point process. In such ensembles, the limit distribution of the k-th largest eigenvalue can be expressed either in terms of a Fredholm determinant or in terms of the Tracy-Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near-extreme eigenvalues, such as the gap between the k-th and the l-th largest eigenvalue, are given in terms of Fredholm determinants with several discontinuities. I will show that these Fredholm determinants have simple expressions in terms of solutions of systems of coupled Painlevé II equations, which are traveling wave reductions of the vector non-linear Schrödinger equation. The talk will be based on joint work in progress with Antoine Doeraene.