The quasi-invariance property for the Gamma kernel determinantal measure

TL;DR

This paper proves that the Gamma kernel determinantal measure remains quasi-invariant under finite permutations of lattice points, which is significant for understanding infinite particle stochastic dynamics.

Contribution

It establishes the quasi-invariance property of the Gamma kernel determinantal measure, extending its applicability to models of infinite particle stochastic dynamics.

Findings

The measure is quasi-invariant under finitary permutations.

Supports analysis of infinite particle stochastic dynamics.

Builds on previous work relating to random partitions.

Abstract

The Gamma kernel is a projection kernel of the form (A(x)B(y)-B(x)A(y))/(x-y), where A and B are certain functions on the one-dimensional lattice expressed through Euler's Gamma function. The Gamma kernel depends on two continuous parameters; its principal minors serve as the correlation functions of a determinantal probability measure P defined on the space of infinite point configurations on the lattice. As was shown earlier (Borodin and Olshanski, Advances in Math. 194 (2005), 141-202; arXiv:math-ph/0305043), P describes the asymptotics of certain ensembles of random partitions in a limit regime. Theorem: The determinantal measure P is quasi-invariant with respect to finitary permutations of the nodes of the lattice. This result is motivated by an application to a model of infinite particle stochastic dynamics.