Quasi-symmetries of determinantal point processes

Bufetov, Alexander I.

doi:10.1214/17-aop1198

Cited by 34 publications

(79 citation statements)

References 34 publications

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“…In [44], the desired quasi-invariance property was established for a particular family of determinantal measures originated from asymptotic representation theory. Then Bufetov [22,Theorem 1.6] showed, by a different method, that this property is not an exceptional phenomenon -it holds for a broad class of measures with projection kernels. Thus, our key condition seems to be reasonable and not too restrictive.…”

Section: Introductionmentioning

confidence: 99%

Determinantal Point Processes and Fermion Quasifree States

Olshanski

2020

Commun. Math. Phys.

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Determinantal point processes are characterized by a special structural property of the correlation functions: they are given by minors of a correlation kernel. However, unlike the correlation functions themselves, this kernel is not defined intrinsically, and the same determinantal process can be generated by many different kernels. The non-uniqueness of a correlation kernel causes difficulties in studying determinantal processes.We propose a formalism which allows to find a distinguished correlation kernel under certain additional assumptions. The idea is to exploit a connection between determinantal processes and quasifree states on CAR, the algebra of canonical anticommutation relations.We prove that the formalism applies to discrete N -point orthogonal polynomial ensembles and to some of their large-N limits including the discrete sine process and the determinantal processes with the discrete Hermite, Laguerre, and Jacobi kernels investigated by Borodin and the author in [16]. As an application we resolve the equivalence/disjointness dichotomy for some of those processes.

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Section: Introductionmentioning

confidence: 99%

Determinantal Point Processes and Fermion Quasifree States

Olshanski

2020

Commun. Math. Phys.

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“…This corollary implies in particular that point (2) in Assumption 1 in [5] can be omitted in the continuous case, that is, it follows from the other assumptions of the main Theorem 1.4 in that paper.…”

Section: Division Properties and Analyticitymentioning

confidence: 74%

“…These are, for example, the sine-kernel [7], Bessel kernel [16], Airy kernel [15], gamma kernel [4,12], discrete sine-kernel [3] and discrete Bessel kernel [3,9]. If an integrable kernel K induces the operator of orthogonal projection onto a closed subspace H ⊂ L 2 (R), then it is shown in [5] that the subspace L has the following division property:…”

Section: Introductionmentioning

confidence: 99%

“…The division property (2) lies at the centre of the proof in [5] of the quasi-invariance of the corresponding processes under the group of compactly supported diffeomorphisms of R, an analogue of the Gibbs property, cf. [14], for determinantal point processes.…”

Section: Introductionmentioning

confidence: 99%

“…The strong division property characterizes the important class of regular de Branges spaces. Recall the following characterization of de Branges spaces: for any de Branges space H there exists a Hermite-Biehler function E such that H coincides with the set of entire functions f such that f/E and f * /E belong to the Hardy class H 2 (C + ), and the identity f H = f/E L 2 (R) holds for all f ∈ H. It is then natural to search for a description of the reproducing kernel Hilbert subspaces in L 2 (R, μ) subject to axiomatic conditions from [5] that ensure quasi-invariance of the corresponding determinantal processes in terms of functional parameter(s) in the spirit of the de Branges theorem just mentioned. In the case when μ is the Lebesgue measure on R such a characterization immediately follows from the elementary theory of the shift operator, see the example in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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Division subspaces and integrable kernels

Bufetov

Romanov

2018

Bull. London Math. Soc.

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Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles

Gangopadhyay,

Ghosh,

Tan

2023

Comm Pure Appl Math

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Gibbsian structure in random point fields has been a classical tool for studying their spatial properties. However, exact Gibbs property is available only in a relatively limited class of models, and it does not adequately address many random fields with a strongly dependent spatial structure. In this work, we provide a very general framework for approximate Gibbsian structure for strongly correlated random point fields, including those with a highly singular spatial structure. These include processes that exhibit strong spatial rigidity, in particular, a certain one‐parameter family of analytic Gaussian zero point fields, namely the α‐GAFs, that are known to demonstrate a wide range of such spatial behavior. Our framework entails conditions that may be verified via finite particle approximations to the process, a phenomenon that we call an approximate Gibbs property. We show that these enable one to compare the spatial conditional measures in the infinite volume limit with Gibbs‐type densities supported on appropriate singular manifolds, a phenomenon we refer to as a generalized Gibbs property. Our work provides a general mechanism to rigorously understand the limiting behavior of spatial conditioning in strongly correlated point processes with growing system size. We demonstrate the scope and versatility of our approach by showing that a generalized Gibbs property holds with a logarithmic pair potential for the α‐GAFs for any value of α. In this vein, we settle in the affirmative an open question regarding the existence of point processes with any specified level of rigidity. In particular, for the α‐GAF zero process, we establish the level of rigidity to be exactly , a fortiori demonstrating the phenomenon of spatial tolerance subject to the local conservation of moments. For such processes involving complex, many‐body interactions, our results imply that the local behavior of the random points still exhibits 2D Coulomb‐type repulsion in the short range. Our techniques can be leveraged to estimate the relative energies of configurations under local perturbations, with possible implications for dynamics and stochastic geometry on strongly correlated random point fields.

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Quasi-symmetries of determinantal point processes

Cited by 34 publications

References 34 publications

Determinantal Point Processes and Fermion Quasifree States

Determinantal Point Processes and Fermion Quasifree States

Division subspaces and integrable kernels

Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles

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