Point Processes and the Infinite Symmetric Group Part Iii: Fermion Point Processes

In Part I (G. Olshanski) and Part II (A. Borodin) we developed an approach to certain probability distributions on the Thoma simplex. The latter has infinite dimension and is a kind of dual object for the infinite symmetric group. Our approach is based on studying the correlation functions of certain related point stochastic processes. In the present paper we consider the so–called tail point processes which describe the limit behavior of the Thoma parameters (coordinates on the Thoma simplex) with large numbers. The tail processes turn out to be stationary processes on the real line. Their correlation functions have determinantal form with a kernel which generalizes the well–known sine kernel arising in random matrix theory. Our second result is a law of large numbers for the Thoma parameters. We also produce Sturm– Liouville operators commuting with the Whittaker kernel introduced in Part II and with the generalized sine kernel.