Stochastic Airy semigroup through tridiagonal matrices
Name
1601.06800.pdf
Description
Submitted version
Size
645.08 KB
Format
Adobe PDF
Checksum (MD5)
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Author(s)
Gorin, Vadim
Shkolnikov, Mykhaylo
Date Issued
June 2018
Journal
The Annals of Probability
Publisher
Institute of Mathematical Statistics
Citation
Gorin, Vadim and Shkolnikov, Mykhaylo. "Stochastic Airy semigroup through tridiagonal matrices." Annals of Probability, 46, no.4, (2018): 2287--2344 © Institute of Mathematical Statistics, 2018.
Version
Original manuscript
Abstract
We determine the operator limit for large powers of random symmetric 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 β process, which describes the largest eigenvalues in the β ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Edelman-Sutton and Ramirez-Rider-Virag. 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. Keywords: Airy point process; Brownian bridge; Brownian excursion; Dumitriu–Edelman model; Feynman–Kac formula; Gaussian beta ensemble; intersection local time; moment method; path transformation; quantile transform; random matrix soft edge; random walk bridge; stochastic Airy operator; strong invariance principle; trace formula; Vervaat transform
Subjects
Statistics, Probability and Uncertainty
Statistics and Probability
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1214/17-aop1229
