# Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions

## TL;DR

This paper investigates the edge behavior of Gaussian beta ensembles with a spike, deriving new Feynman-Kac formulas for associated stochastic operators using reflected Brownian motion, advancing understanding of eigenvalue fluctuations.

## Contribution

It introduces a novel analysis of spiked Gaussian beta ensembles at the edge, connecting finite-dimensional models to infinite-dimensional stochastic semigroups and operators.

## Key findings

- Derived Feynman-Kac formulas for spiked stochastic Airy operators
- Established a distributional identity for reflected Brownian bridges
- Linked finite-dimensional eigenvalue fluctuations to infinite-dimensional stochastic semigroups

## 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 beta=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 [13] as an extreme case. Our analysis also provides Feynman-Kac formulas for the spiked stochastic Airy operators, introduced by Bloemendal and Virag [6]. 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.