Stochastic Airy semigroup through tridiagonal matrices

TL;DR

This paper establishes the operator limit for large powers of random tridiagonal matrices, linking it to the stochastic Airy process and providing new probabilistic representations and formulas related to random matrix theory.

Contribution

It introduces a novel operator limit for large powers of random tridiagonal matrices and connects it to the stochastic Airy process through new functional representations.

Findings

Derived a new expression for the Laplace transform of the Airy_beta process.

Established a Feynman-Kac formula for the stochastic Airy operator.

Discovered a Gaussian fluctuation related to Brownian excursions.

Abstract

We determine the operator limit for large powers of random 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 Ram\'{i}rez, Rider, and Vir\'{a}g. 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.