Evolution of the Stochastic Airy eigenvalues under a changing boundary

TL;DR

This paper investigates how the eigenvalues of the stochastic Airy operator evolve as the boundary shifts, revealing differentiability and stationarity properties of the resulting point processes.

Contribution

It introduces a coupled family of eigenvalue point processes parameterized by boundary position and proves their differentiability and stationarity properties.

Findings

Eigenvalues form a differentiable process in boundary parameter t.

Recentered eigenvalue process is stationary.

Analogy to GUE minor process in tridiagonal models.

Abstract

The Airy$_\beta$ point process, originally introduced by Ram\'irez, Rider, and Vir\'ag, is defined as the spectrum of the stochastic Airy operator $\mathcal{H}_\beta$ acting on a subspace of $L^2[0,\infty)$ with Dirichlet boundary condition. In this paper we study the coupled family of point processes defined as the eigenvalues of $\mathcal{H}_\beta$ acting on a subspace of $L^2[t,\infty)$. These point processes are coupled through the Brownian term of $\mathcal{H}_\beta$. We show that these point processes as a function of $t$ are differentiable with explicitly computable derivative. Moreover when recentered by $t$ the resulting point process is stationary. This process can also be viewed as an analogue to the 'GUE minor process' in the tridiagonal setting.