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
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
SN - 0246-0203
IS - 3
ER -