## Abstract

We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases β = 1, 2, 4, this corresponds to studying the fluctuations of the largest eigenvalues of additive rank one perturbations of the GOE/GUE/GSE random matrices. In the infinite-dimensional limit, we arrive at a one-parameter family of random Feynman-Kac type semigroups, which features the stochastic Airy semigroup of Gorin and Shkolnikov (Ann. Probab. 46 (2018) 2287-2344) as an extreme case. Our analysis also provides Feynman-Kac formulas for the spiked stochastic Airy operators, introduced by Bloemendal and Virag (Probab. Theory Related Fields 156 (2013) 795-825). The Feynman. Kac formulas involve functionals of a reflected Brownian motion and its local times, thus, allowing to study the limiting operators by tools of stochastic analysis. We derive a first result in this direction by obtaining a new distributional identity for a reflected Brownian bridge conditioned on its local time at zero. A key feature of our proof consists of a novel strong invariance result for certain non-negative random walks and their occupation times that is based on the Skorokhod reflection map.

Original language | English (US) |
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Pages (from-to) | 1402-1438 |

Number of pages | 37 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 55 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2019 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Keywords

- Beta ensembles
- Feynman-Kac formulas
- Local times
- Low rank perturbations
- Moments method
- Operator limits
- Path transformations
- Random tridiagonal matrices
- Reflected Brownian motions
- Skorokhod map
- Stochastic Airy semigroups
- Strong invariance principles