Evolution of the Stochastic Airy eigenvalues under a changing boundary

The Airy$_β$ point process, originally introduced by Ramírez, Rider, and Virág, is defined as the spectrum of the stochastic Airy operator $\mathcal{H}_β$ 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 $\mathcal{H}_β$ acting on a subspace of $L^2[t,\infty)$. These point processes are coupled through the Brownian term of $\mathcal{H}_β$. 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.

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