The quasi-invariance property for the Gamma kernel determinantal measure

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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 Γ-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 [A. Borodin, G. Olshanski, Adv. 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.

Original languageEnglish
Pages (from-to)2305-2350
Number of pages46
JournalAdvances in Mathematics
Issue number3
Publication statusPublished - 15 Feb 2011
Externally publishedYes


  • Correlation kernels
  • Determinantal point processes
  • Gamma kernel
  • Quasi-invariant measures
  • Radon-Nikodỳm derivative


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