### 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 language | English (US) |
---|---|

Pages (from-to) | 2287-2344 |

Number of pages | 58 |

Journal | Annals of Probability |

Volume | 46 |

Issue number | 4 |

DOIs | |

State | Published - 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

## Fingerprint Dive into the research topics of 'Stochastic Airy semigroup through tridiagonal matrices'. Together they form a unique fingerprint.

## Cite this

Gorin, V., & Shkolnikov, M. (2018). Stochastic Airy semigroup through tridiagonal matrices.

*Annals of Probability*,*46*(4), 2287-2344. https://doi.org/10.1214/17-AOP1229