## The Annals of Probability

### 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$_{\beta}$ 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.

#### Article information

Source
Ann. Probab., Volume 46, Number 4 (2018), 2287-2344.

Dates
Revised: August 2017
First available in Project Euclid: 13 June 2018

https://projecteuclid.org/euclid.aop/1528876829

Digital Object Identifier
doi:10.1214/17-AOP1229

Mathematical Reviews number (MathSciNet)
MR3813993

Zentralblatt MATH identifier
06919026

#### Citation

Gorin, Vadim; Shkolnikov, Mykhaylo. Stochastic Airy semigroup through tridiagonal matrices. Ann. Probab. 46 (2018), no. 4, 2287--2344. doi:10.1214/17-AOP1229. https://projecteuclid.org/euclid.aop/1528876829

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#### Supplemental materials

• Global asymptotics. We analyze the asymptotics of global linear statistics of the spectrum of symmetric tridiagonal matrices and prove a cental limit theorem for them.
• Coupling with Brownian bridge local times. We construct a coupling with the local times of the Brownian bridge and prove Proposition 4.1.