TracyWidom GUE law and symplectic invariants
Abstract
We establish the relation between two objects: an integrable system related to Painleve II equation, and the symplectic invariants of a certain plane curve \Sigma_{TW} describing the average eigenvalue density of a random hermitian matrix spectrum near a hard edge (a bound for its maximal eigenvalue). This explains directly how the TracyWidow law F_{GUE}, governing the distribution of the maximal eigenvalue in hermitian random matrices, can also be recovered from symplectic invariants.
 Publication:

arXiv eprints
 Pub Date:
 November 2010
 arXiv:
 arXiv:1011.1418
 Bibcode:
 2010arXiv1011.1418B
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematical Physics;
 30Exx;
 34E05;
 33E17;
 60F10
 EPrint:
 pdfLatex, 36 pages, 1 figure. v2: typos corrected, resectioning, a reference added