Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Interlaced processes on the circle

Anthony P. Metcalfe, Neil O’Connell, and Jon Warren

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Abstract

When two Markov operators commute, it suggests that we can couple two copies of one of the corresponding processes. We explicitly construct a number of couplings of this type for a commuting family of Markov processes on the set of conjugacy classes of the unitary group, using a dynamical rule inspired by the RSK algorithm. Our motivation for doing this is to develop a parallel programme, on the circle, to some recently discovered connections in random matrix theory between reflected and conditioned systems of particles on the line. One of the Markov chains we consider gives rise to a family of Gibbs measures on “bead configurations” on the infinite cylinder. We show that these measures have determinantal structure and compute the corresponding space–time correlation kernel.

Résumé

Quand deux opérateurs de Markov commutent, cela suggère que nous pouvons coupler deux copies d’un des processus correspondants. Nous construisons explicitement un certain nombre de couplages de ce type pour une famille de processus de Markov qui commutent sur l’ensemble des classes de conjugaison du groupe unitaire. Nous utilisons, à cette fin, une règle dynamique inspirée par l’algorithme RSK. Notre motivation est de développer un programme parallèle sur le cercle, pour des connections récemment mises à jour dans la théorie des matrices aléatoires entre des systèmes de particules réfléchies et conditionnées sur la droite. Une des chaînes de Markov que nous considérons donne lieu à une famille de mesures de Gibbs sur des configurations de perles sur le cylindre infini. Nous prouvons que ces mesures ont la structure déterminantale et calculons le noyau de corrélation espace-temps correspondant.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 4 (2009), 1165-1184.

Dates
First available in Project Euclid: 6 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1257529898

Digital Object Identifier
doi:10.1214/08-AIHP305

Mathematical Reviews number (MathSciNet)
MR2572170

Zentralblatt MATH identifier
1218.60075

Subjects
Primary: 60J99: None of the above, but in this section 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 82B21: Continuum models (systems of particles, etc.)
Secondary: 05E10: Combinatorial aspects of representation theory [See also 20C30]

Keywords
Random matrices RSK correspondence coupling interlacing rank 1 perturbation random reflection Pitman’s theorem reflected Brownian motion Brownian motion in an alcove bead model on a cylinder determinantal point process random tiling dimer configuration

Citation

Metcalfe, Anthony P.; O’Connell, Neil; Warren, Jon. Interlaced processes on the circle. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 4, 1165--1184. doi:10.1214/08-AIHP305. https://projecteuclid.org/euclid.aihp/1257529898


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