Infinite-dimensional diffusions as limits of random walks on partitions

Abstract

The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the z-measures, which depend on the complex parameter z. The z-measures live on the Thoma simplex, an infinite-dimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the z-measures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtained from certain Markov chains on partitions of natural numbers n in a scaling limit as n goes to infinity. These Markov chains arise in a natural way, due to the approximation of the infinite symmetric group by the increasing chain of the finite symmetric groups. Each z-measure P_z serves as a unique invariant distribution for the corresponding diffusion process, and the process is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so that the process is reversible. We describe the spectrum of its generator and compute the associated (pre)Dirichlet form.

AMSTex, 33 pages. Version 2: minor changes, typos corrected, to appear in Prob. Theor. Rel. Fields

References in corpus (6)

• Shifted Schur Functions
• Distributions on partitions, point processes, and the hypergeometric kernel
• Harmonic analysis on the infinite symmetric group
• Z-Measures on partitions, Robinson-Schensted-Knuth correspondence, and beta=2 random matrix ensembles
• SL(2) and z-measures
• A Two-Parameter Family of Infinite-Dimensional Diffusions in the Kingman Simplex

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• Non-colliding Jacobi processes as limits of Markov chains on Gelfand-Tsetlin graph
• Random Strict Partitions and Determinantal Point Processes
• Pfaffian Stochastic Dynamics of Strict Partitions
• Laguerre and Meixner orthogonal bases in the algebra of symmetric functions
• Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles
• A sharp analysis of the mixing time for random walk on rooted trees
• Coherent systems of probability measures on graphs for representations of free Frobenius towers
• Transition Density of an Infinite-dimensional diffusion with the Jack Parameter