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arXiv · 0706.1034

Infinite-dimensional diffusions as limits of random walks on partitions

Alexei Borodin·Grigori Olshanski
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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.

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Alexei Borodin, Grigori Olshanski. 2008-02-24. Infinite-dimensional diffusions as limits of random walks on partitions. https://doi.org/10.1007/s00440-008-0148-8

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