Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Pierre Yves Gaudreau Lamarre and Mykhaylo Shkolnikov

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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 $\beta =1,2,4$, 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 Virág (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.


Nous accédons à l’extrémité du spectre des ensembles bêta gaussiens avec perturbation de rang un par l’entremise de grandes puissances des matrices tridiagonales qui y sont associées. Pour les valeurs traditionnelles $\beta =1,2,4$, ceci correspond à l’étude des fluctuations des valeurs propres maximales des matrices aléatoires GOE/GUE/GSE assujetties à une perturbation additive de rang un. En dimensions infinies, nos résultats nous mènent vers une famille de semi-groupes de type Feynman–Kac qui, dans un cas extrême, correspond au stochastic Airy semigroup introduit par Gorin et Shkolnikov (Ann. Probab. 46 (2018) 2287–2344). De plus, nos résultats ont pour corollaire des formules de Feynman–Kac pour les spiked stochastic Airy operators introduits par Bloemendal et Virág (Probab. Theory Related Fields 156 (2013) 795–825). Ces formules sont exprimées à l’aide de certaines fonctionnelles du mouvement brownien réfléchi et de ses temps locaux. Ce faisant, les opérateurs en question peuvent être étudiés à l’aide du calcul stochastique. Nous obtenons un premier résultat dans cette lignée en démontrant une nouvelle identité décrivant la distribution du mouvement brownien réfléchi ayant été conditionné sur son temps local à zéro. La principale innovation de notre démonstration consiste en la preuve d’un nouveau résultat sur l’approximation forte du mouvement brownien réfléchi et de son temps local par une marche aléatoire non négative en utilisant la méthode de réflexion de Skorokhod.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1402-1438.

Received: 2 October 2017
Revised: 29 April 2018
Accepted: 27 July 2018
First available in Project Euclid: 25 September 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60H25: Random operators and equations [See also 47B80] 47D08: Schrödinger and Feynman-Kac semigroups 60J55: Local time and additive functionals

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


Lamarre, Pierre Yves Gaudreau; Shkolnikov, Mykhaylo. Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1402--1438. doi:10.1214/18-AIHP923.

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