Tracy-Widom 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 Tracy-Widow law F_{GUE}, governing the distribution of the maximal eigenvalue in hermitian random matrices, can also be recovered from symplectic invariants.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2010
- DOI:
- 10.48550/arXiv.1011.1418
- arXiv:
- arXiv:1011.1418
- Bibcode:
- 2010arXiv1011.1418B
- Keywords:
-
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- Mathematical Physics;
- 30Exx;
- 34E05;
- 33E17;
- 60F10
- E-Print:
- pdfLatex, 36 pages, 1 figure. v2: typos corrected, re-sectioning, a reference added