TY - JOUR

T1 - Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions

AU - Gaudreau Lamarre, Pierre Yves

AU - Shkolnikov, Mykhaylo

N1 - Funding Information:
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 1P. Y. Gaudreau Lamarre is partially supported by an NSERC Doctoral Fellowship and a Gordon Y. S. Wu Fellowship. 2M. Shkolnikov is partially supported by the NSF Grant DMS-1506290.
Funding Information:
P. Y. Gaudreau Lamarre is partially supported by an NSERC Doctoral Fellowship and a Gordon Y. S. Wu Fellowship. M. Shkolnikov is partially supported by the NSF Grant DMS-1506290.
Publisher Copyright:
© 2019 Association des Publications de l'Institut Henri Poincaré.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Beta ensembles

KW - Feynman-Kac formulas

KW - Local times

KW - Low rank perturbations

KW - Moments method

KW - Operator limits

KW - Path transformations

KW - Random tridiagonal matrices

KW - Reflected Brownian motions

KW - Skorokhod map

KW - Stochastic Airy semigroups

KW - Strong invariance principles

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U2 - 10.1214/18-AIHP923

DO - 10.1214/18-AIHP923

M3 - Article

AN - SCOPUS:85074195392

SN - 0246-0203

VL - 55

SP - 1402

EP - 1438

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 3

ER -