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Evolution of the Stochastic Airy eigenvalues under a changing boundary
- Publication Year :
- 2020
-
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.<br />Comment: 17 pages, 3 figures
- Subjects :
- Mathematics - Probability
60B20
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2002.12191
- Document Type :
- Working Paper