Stochastic Airy semigroup through tridiagonal matrices

Vadim Gorin, Mykhaylo Shkolnikov

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

We determine the operator limit for large powers of random symmetric tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy β process, which describes the largest eigenvalues in the β ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Edelman-Sutton and Ramirez-Rider-Virag. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable.

Original languageEnglish (US)
Pages (from-to)2287-2344
Number of pages58
JournalAnnals of Probability
Volume46
Issue number4
DOIs
StatePublished - Jan 1 2018

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Airy point process
  • Brownian bridge
  • Brownian excursion
  • Dumitriu- Edelman model
  • Feynman-Kac formula
  • Gaussian beta ensemble
  • Intersection local time
  • Moment method
  • Path transformation
  • Quantile transform
  • Random matrix soft edge
  • Random walk bridge
  • Stochastic Airy operator
  • Strong invariance principle
  • Trace formula
  • Vervaat transform

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