Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel

Abstract. For a determinantal point process induced by the reproducing kernel of the weighted Bergman space A 2 (U, ω) over a domain U ⊂ C d , we establish the mutual absolute continuity of reduced Palm measures of any order provided that the domain U contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the H ∞ (U )-module structure of A 2 (U, ω). A corollary is the quasi-invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of U .

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“…is a gamma kernel measure and p ∈ Z ′ . Note that M (z,z ′ ) (p) and M (z,z ′ ) are mutually singular, see [14,Proposition 4.3].…”

Section: 5mentioning

confidence: 99%

“…Let M (z,z ′ ) be the corresponding gamma kernel measure and M (z,z ′ ) (p) be its reduced Palm measure at p (it is well defined by virtue of Corollary 7.6). Note that the measures M (z,z ′ ) and M (z,z ′ ) (p) are mutually singular, see [14].…”

Section: A Hierarchy Of Palm Measuresmentioning

confidence: 99%

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

Bufetov

Fan

Qiu

2017

Probab. Theory Relat. Fields

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“…The difference ϕ (R,T ) (x) − ϕ (R,T ) (y) is zero on {(x, y)|R > y ≥ x}, therefore it is sufficient to consider the two other domains D >R and D <R . First we restate Proposition 1.1 from [2] in our case as follows.…”

Section: Preliminary Propositionsmentioning

confidence: 99%

“…We next give a general sufficient condition for the number rigidity of a stationary point process convenient for working with stationary Pfaffian processes. Let P be a stationary point process on R admitting the first and the second correlation functions ρ (1) P and ρ (2) P . The first correlation function is a constant, and we set ρ = ρ (1) P (x).…”

Section: Pfaffian Point Processesmentioning

confidence: 99%

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On Number Rigidity for Pfaffian Point Processes

Bufetov¹,

Nikitin²,

Qiu³

2019

MMJ

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Our first result states that the orthogonal and symplectic Bessel processes are rigid in the sense of Ghosh and Peres. Our argument in the Bessel case proceeds by an estimate of the variance of additive statistics in the spirit of Ghosh and Peres. Second, a sufficient condition for number rigidity of stationary Pfaffian processes, relying on the Kolmogorov criterion for interpolation of stationary processes and applicable, in particular, to pfaffian sine-processes, is given in terms of the asymptotics of the spectral measure for additive statistics.

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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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Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel

Cited by 55 publications

References 23 publications

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

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

On Number Rigidity for Pfaffian Point Processes

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

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