Abstract
Infinite determinantal measures introduced in this note areinductive limits of determinantal measures on an exhaustingfamily of subsets of the phase space. Alternatively,an infinite determinantal measure can be describedas a product of a determinantal point processand a convergent, but not integrable, multiplicative functional.   Theorem 4.1, the main result announced in this note, gives an explicit description forthe ergodic decomposition of infinite Pickrell measures on thespaces of infinite complex matrices in terms of infinitedeterminantal measures obtained by finite-rank perturbations ofBessel point processes.
Highlights
In Theorem 2 of Section 4, infinite determinantal measures will be seen to arise in the ergodic decomposition of infinite unitarily-invariant measures on spaces of infinite complex matrices
A configuration on the phase space E is an unordered collection of points of E, possibly with multiplicities; the main assumption is that a bounded subset of E contain only finitely many particles of a given configuration
To a function g on E assign its multiplicative functional Ψg on the space of configurations: the functional Ψg is obtained by multiplying the values of g over all particles of a configuration (see (5))
Summary
If a subspace H ⊂ L2,loc(E, μ) and a Borel subset E0 ⊂ E satisfy Assumption 1, there exists a σ-finite Borel measure B on Conf(E) such that (1) B-almost every configuration has at most finitely many particles outside of E0; (2) for any bounded Borel (possibly empty) subset B ⊂ E \ E0 we have 0 < B(Conf(E; E0 ∪ B)) < +∞ and. The main result of the paper, Theorem 2 , says that the ergodic decomposition of infinite Pickrell measures is induced by infinite determinantal measures obtained as an explicitly given finite-rank perturbation of the Bessel point processes occurring in the ergodic decomposition of finite Pickrell measures. Let a subspace H ⊂ L2,loc(E, μ) and a Borel subset E0 ⊂ E satisfy Assumption 1, and let g : E → Corollary 1 means that, as n grows, the induced processes of our determinantal measure on subsets Conf(E; E0 ∪ Bn) converge to the “unperturbed” determinantal point process PQ
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