Xavier Navand

Fredholm Determinants and Random Matrices

First, random matrix theory will be introduced, summarizing the main results necessary to discuss the cumulative distribution for the largest eigenvalue of the Gaussian Unitary Ensemble. When the size of the matrix becomes infinite, this limiting distribution becomes a Fredholm determinant which can be evaluated by a functional formula, yielding the Tracy-Widom distribution. Then, a generalized setup will be described and motivated. Finally, our recent result about these generalized considerations will be presented as a deformation of the Tracy-Widom distribution, and relations to integrable systems will be briefly discussed.