Interlaced processes on the circle
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 1165-1184.

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.

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.

DOI : 10.1214/08-AIHP305
Classification : 60J99, 60B15, 82B21, 05E10
Mots-clés : 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 
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Metcalfe, Anthony P.; O’Connell, Neil; Warren, Jon. Interlaced processes on the circle. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 1165-1184. doi : 10.1214/08-AIHP305. http://archive.numdam.org/articles/10.1214/08-AIHP305/

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