Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions

Pierre Yves Gaudreau Lamarre, Mykhaylo Shkolnikov

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases β = 1, 2, 4, this corresponds to studying the fluctuations of the largest eigenvalues of additive rank one perturbations of the GOE/GUE/GSE random matrices. In the infinite-dimensional limit, we arrive at a one-parameter family of random Feynman-Kac type semigroups, which features the stochastic Airy semigroup of Gorin and Shkolnikov (Ann. Probab. 46 (2018) 2287-2344) as an extreme case. Our analysis also provides Feynman-Kac formulas for the spiked stochastic Airy operators, introduced by Bloemendal and Virag (Probab. Theory Related Fields 156 (2013) 795-825). The Feynman. Kac formulas involve functionals of a reflected Brownian motion and its local times, thus, allowing to study the limiting operators by tools of stochastic analysis. We derive a first result in this direction by obtaining a new distributional identity for a reflected Brownian bridge conditioned on its local time at zero. A key feature of our proof consists of a novel strong invariance result for certain non-negative random walks and their occupation times that is based on the Skorokhod reflection map.

Original languageEnglish (US)
Pages (from-to)1402-1438
Number of pages37
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume55
Issue number3
DOIs
StatePublished - Jan 1 2019

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Beta ensembles
  • Feynman-Kac formulas
  • Local times
  • Low rank perturbations
  • Moments method
  • Operator limits
  • Path transformations
  • Random tridiagonal matrices
  • Reflected Brownian motions
  • Skorokhod map
  • Stochastic Airy semigroups
  • Strong invariance principles

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