@article{1ae42f114dda4cdab5402f6cf28755b8,
title = "Stochastic Airy semigroup through tridiagonal matrices",
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.",
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",
author = "Vadim Gorin and Mykhaylo Shkolnikov",
note = "Funding Information: 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2288 2. Setup and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2289 3. Combinatorics of high powers of symmetric tridiagonal matrices . . . . . . . . . . . . . . . 2297 4. Toward a rigorous proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2300 4.1. Convergence of random walk bridges and their local times . . . . . . . . . . . . . . . 2300 4.2. Leading order terms in Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2313 4.3. Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2328 5. Properties of the stochastic Airy semigroup I . . . . . . . . . . . . . . . . . . . . . . . . . . 2330 6. Convergence of extreme eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2336 7. Properties of the stochastic Airy semigroup II . . . . . . . . . . . . . . . . . . . . . . . . . 2339 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2341 Supplementary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2341 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2341 Received June 2016; revised August 2017. 1Supported in part by NSF Grants DMS-1407562, DMS-1664619 and by the Sloan Research Fellowship. 2Supported in part by NSF Grant DMS-1506290. MSC2010 subject classifications. Primary 60B20, 60H25; secondary 47D08, 60G55, 60J55. Key words and phrases. 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. Publisher Copyright: {\textcopyright} Institute of Mathematical Statistics, 2018.",
year = "2018",
doi = "10.1214/17-AOP1229",
language = "English (US)",
volume = "46",
pages = "2287--2344",
journal = "Annals of Probability",
issn = "0091-1798",
publisher = "Institute of Mathematical Statistics",
number = "4",
}