Multiplicative functionals of determinantal processes

Equivalence of Palm measures for determinantal point processes governed by Bergman kernels

“…, where L is a closed subspace of ℓ 2 (X), K be the operator of orthogonal projection onto L, and K(x, y) be the matrix of K. The next theorem extends Lemma 6.3 to a wider class of functions a(x). Note that this theorem is a particular case of Proposition 4.2 in [13], which, in turn, is a generalization of Proposition 2.1 in Bufetov [16], previously announced in [15].…”

Quasi-symmetries of determinantal point processes

“…At the centre of the argument lies the result of [6] that can informally be summarized as follows: a determinantal measure times a multiplicative functional is, after normalization, again a determinantal measure. More precisely, let g be a positive function on E bounded away from 0 and ∞, and let Π be an operator of orthogonal projection in L 2 (E) onto a closed subspace L. Let Π g be the operator of orthogonal projection onto the subspace √ gL.…”

“…Olshanski [21] proves the quasi-invariance of the Gamma-kernel process by a limit transition from finite-dimensional approximations. The argument in this paper is direct: first, it is shown that the Palm subspaces corresponding to conditioning at points p and q are taken one to the other by multiplication by the function (x − p)/(x − q); after which, the proof is completed using a general result of [5], [6] that multiplying the range of the projection operator Π inducing a determinantal measure P Π by a function g, corresponds, under certain additional assumptions, to taking the product of the determinantal measure P Π by the multiplicative functional Ψ g induced by the function g. The key technical step is the regularization of divergent multiplicative functionals. This paper is devoted to determinantal point processes governed by orthogonal projections; in the case of contractions, quasi-invariance is due to Camilier and Decreusefond [8]; note that in their case the Radon-Nikodym derivative exhibits a much more sensitive dependence on the specific kernel.…”

A rigidity property of superpositions involving determinantal processes

“…Here the Fredholm determinant det(1+ϕK1 supp e (ϕ) ) is well-defined since the locally trace class implies that ϕK1 supp e (ϕ) is trace class. The reader is referred to Simon [11] for more details on Fredholm determinants and to [13, Theorem 2], [10, Theorem 1.2] or [2] and [3, formula (16)] for more details on the formula (1.3).…”