The Annals of Probability

Stochastic Airy semigroup through tridiagonal matrices

Vadim Gorin and Mykhaylo Shkolnikov

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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

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

Received: June 2016
Revised: August 2017
First available in Project Euclid: 13 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60H25: Random operators and equations [See also 47B80]
Secondary: 47D08: Schrödinger and Feynman-Kac semigroups 60G55: Point processes 60J55: Local time and additive functionals

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


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

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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.