Quasi-Symmetries for Pfaffian Point Processes
Alexander Bufetov (Aix-Marseille Université)
For a given domain E in Euclidean space, we denote Conf(E) the set of locally finite configurations on E and denote Diffc(E) the group of compactly supported C1-diffeomorphisms of E. The tautological action of Diffc(E) on E induces a natural action of Diffc(E) on Conf(E). We say a point process P on E (i.e., a Borel probability measure P on Conf(E)) is Diffc(E)-quasi-symmetric, if the Diffc(E)-action preserves the measure class of P. In the setting of determinantal point processes, the quasi-symmetries have been obtained by Bufetov for determinantal point processes on the real line R induced by integrable kernels, including Dyson-sine process, Bessel processes, Airy process etc and have also been obtained by Bufetov-Qiu for determinantal point processes on the complex plane C or the unit disk D ⊂ C associated with a class of Hilbert spaces of holomorphic functions. Note that in all these cases, the associated Radon-Nikodym derivatives can be expressed as certain regularized multiplicative functionals. Pfaffian point processes arise naturally in many similar situations where determinantal point processes arise and they share certain similarity as determinantal point processes, for instance, both of them possess the repulsive nature and their Laplace transforms are given by Fredholm determinants or Fredholm Pfaffians. A natural problem that we would like to investigate is whether certain important Pfaffian point processes, for instance, the classical sine1 and sine4 point processes on the real line R, also possess the quasi-symmetries?
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement N°647113)