Point processes and the infinite symmetric group. Part I: The general formalism and the density function

TL;DR

This paper introduces a probabilistic framework for analyzing measures on the Thoma simplex related to the infinite symmetric group, linking correlation functions to multidimensional moment problems and exploring their support and applications.

Contribution

It develops a new interpretation of measures on the Thoma simplex as point processes, relating correlation functions to moment problems, and computes the first correlation function to analyze measure support.

Findings

Correlation functions relate to solutions of multidimensional moment problems.

First correlation function calculation reveals measure support properties.

Framework connects harmonic analysis on infinite groups with point process theory.

Abstract

We study a 2-parametric family of probability measures on an infinite-dimensional simplex (the Thoma simplex). These measures originate in harmonic analysis on the infinite symmetric group (S.Kerov, G.Olshanski and A.Vershik, Comptes Rendus Acad. Sci. Paris I 316 (1993), 773-778). Our approach is to interpret them as probability distributions on a space of point configurations, i.e., as certain point stochastic processes, and to find the correlation functions of these processes. In the present paper we relate the correlation functions to the solutions of certain multidimensional moment problems. Then we calculate the first correlation function which leads to a conclusion about the support of the initial measures. In the appendix, we discuss a parallel but more elementary theory related to the well-known Poisson-Dirichlet distribution. The higher correlation functions are explicitly calculated in the subsequent paper (A.Borodin, math.RT/9804087). In the third part (A.Borodin and G.Olshanski, math.RT/9804088) we discuss some applications and relationships with the random matrix theory. The goal of our work is to understand new phenomena in noncommutative harmonic analysis which arise when the irreducible representations depend on countably many continuous parameters.