From Gumbel to Tracy-Widom

Abstract

The Tracy-Widom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution $\exp (-\exp (-x))$, the Gumbel distribution and the Tracy-Widom distribution. There is a family of determinantal processes whose edge behaviour interpolates between a Poisson process with density $\exp (-x)$ and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the deformed GUE ensemble, $M=M_0+\sqrt{2S}V$, with $M_0$ diagobal with independent elements and $V$ from GUE. Here we do not see a transition from Tracy-Widom to Gumbel, but rather a transition from Tracy-Widom to Gaussian.

29 pages

References in corpus (11)

• Level-Spacing Distributions and the Airy Kernel
• Discrete polynuclear growth and determinantal processes
• Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices
• Random matrices and determinantal processes
• A generalized ensemble of random matrices
• Transitions toward Quantum Chaos: with Supersymmetry from Poisson to Gauss
• Characterization of invariant measures at the leading edge for competing particle systems
• Toeplitz determinants, random growth and determinantal processes
• Fluctuation of maximal particle energy of quantum ideal gas and random partitions
• Between Poisson and GUE statistics: Role of the Breit-Wigner width
• Correlations for the Dyson Brownian motion model with Poisson initial conditions

Cited by in corpus (1)

• On some special directed last-passage percolation models