Open Access
July 2018 Stochastic Airy semigroup through tridiagonal matrices
Vadim Gorin, Mykhaylo Shkolnikov
Ann. Probab. 46(4): 2287-2344 (July 2018). DOI: 10.1214/17-AOP1229


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$_{\beta}$ process, which describes the largest eigenvalues in the $\beta$ 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.


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Vadim Gorin. Mykhaylo Shkolnikov. "Stochastic Airy semigroup through tridiagonal matrices." Ann. Probab. 46 (4) 2287 - 2344, July 2018.


Received: 1 June 2016; Revised: 1 August 2017; Published: July 2018
First available in Project Euclid: 13 June 2018

zbMATH: 06919026
MathSciNet: MR3813993
Digital Object Identifier: 10.1214/17-AOP1229

Primary: 60B20 , 60H25
Secondary: 47D08 , 60G55 , 60J55

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

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 4 • July 2018
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