From Gumbel to Tracy-Widom

2005/10/10 by Kurt Johansson, Johansson, Kurt · 2 citations
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.math/0510181

openalex publication_date 2005/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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=M₀+√(2S) V, with M₀ 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.

Citations

• Between Poisson and GUE Statistics: Role of the Breit–Wigner Width

Cited by

• Extreme events scaling in self-organized critical models
• Large deviations at the edge for 1D gases and tridiagonal random matrices at high temperature

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