PDE's for the Gaussian ensemble with external source and the Pearcey distribution
| dc.creator | Adler, Mark | |
| dc.creator | van Moerbeke, Pierre | |
| dc.date | 2005-09-02 | |
| dc.date.accessioned | 2026-07-07T05:22:55Z | |
| dc.date.available | 2026-07-07T05:22:55Z | |
| dc.description | The present paper studies a Gaussian Hermitian random matrix ensemble with external source, given by a fixed diagonal matrix with two eigenvalues a and -a. As a first result, the probability that the eigenvalues of the ensemble belong to a set satisfies a fourth order PDE with quartic non-linearity; the variables being the eigenvalue a and the boundary points of the set. This equation enables one to find a PDE for the Pearcey distribution. The latter describes the statistics of the eigenvalues near the closure of a gap; i.e., when the support of the equilibrium measure for large size random matrices has a gap, which can be made to close. Precisely, the Gaussian Hermitian random matrix ensemble with external source has this feature. In this work, we show the Pearcey distribution satisfies a a fourth order PDE with cubic non-linearity. The PDE for the finite problem is found by by showing that an appropriate integrable deformation of the random matrix ensemble with external source satisfies the three-component KP equation and Virasoro constraints. | |
| dc.description | 33 pages | |
| dc.identifier | https://arxiv.org/abs/math/0509047 | |
| dc.identifier | http://arxiv.org/abs/math/0509047 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/76244 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.subject | 60J60, 60J65, 60G55; 35Q53, 35Q58 | |
| dc.title | PDE's for the Gaussian ensemble with external source and the Pearcey distribution | |
| dc.type | text |