Central limit theorem for spectral partial Bergman kernels

Zelditch, Steve; Zhou, Peng

doi:10.2140/gt.2019.23.1961

Cited by 24 publications

(44 citation statements)

References 26 publications

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“…The time-scaled Toeplitz Fourier integral operators U N ( t √ N ) are studied in [11,12] for unrelated reasons, and the pointwise and integrated asymptotics of those articles gives Theorem 1.3.…”

Section: Statement Of Results In the Bargmann-fock Settingmentioning

confidence: 99%

“…In this section, we prove Theorem 1.3 as outlined in Section 1.2. The proof is based on results of [11,12] and involves Toeplitz Fourier integral operators acting on holomorphic sections of the standard line bundles O(N) → CP d−1 over the projective space. Since it is our second proof of the main result, we do not provide detailed background on Toeplitz Fourier integral operators and refer to [11,12] for further background and references.…”

Section: Conjugation To the Bargmann-fock Modelmentioning

confidence: 99%

“…The proof is based on results of [11,12] and involves Toeplitz Fourier integral operators acting on holomorphic sections of the standard line bundles O(N) → CP d−1 over the projective space. Since it is our second proof of the main result, we do not provide detailed background on Toeplitz Fourier integral operators and refer to [11,12] for further background and references. Our goal to describe the conjugation to the holomorphic setting and to connect the asymptotics of Theorem 1.3 to those of [11,12].…”

Section: Conjugation To the Bargmann-fock Modelmentioning

confidence: 99%

“…Since it is our second proof of the main result, we do not provide detailed background on Toeplitz Fourier integral operators and refer to [11,12] for further background and references. Our goal to describe the conjugation to the holomorphic setting and to connect the asymptotics of Theorem 1.3 to those of [11,12].…”

Section: Conjugation To the Bargmann-fock Modelmentioning

confidence: 99%

“…After generalizing the averaging technique of Weinstein [10] and Guillemin [5], we prove the result in two ways: (i) using special parametrices for the isotropic propagators U(t) on L 2 (R d ) developed in [3], and (ii) by using an eigespace decomposition for H 0 and conjugating the traces of U(t) on eigenspaces to the Bargmann-Fock representation and then to holomorphic sections of line bundles over the space CP d−1 of Hamilton orbits of H 0 . It turns out that the eigenspace traces of U(t) involve the central limit scaling of Toeplitz propagators studied in [11,12]. We then use the trace asymptotics of [12] to obtain a second proof.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Refined Weyl Law for Homogeneous Perturbations of the Harmonic Oscillator

Doll

Gannot²,

Wunsch³

2018

Commun. Math. Phys.

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Let H = H 0 + P denote the harmonic oscillator on R d perturbed by an isotropic pseudodifferential operator P of order 1 and let U (t) = exp(−itH). We prove a Gutzwiller-Duistermaat-Guillemin type trace formula for Tr U (t). The singularities occur at times t ∈ 2πZ and the coefficients involve the dynamics of the Hamilton flow of the symbol σ(P ) on the space CP d−1 of harmonic oscillator orbits of energy 1. This is a novel kind of sub-principal symbol effect on the trace. We generalize the averaging technique of Weinstein and Guillemin to this order of perturbation, and then present two completely different calculations of Tr U (t). The first proof directly constructs a parametrix of U (t) in the isotropic calculus, following earlier work of Doll-Gannot-Wunsch. The second proof conjugates the trace to the Bargmann-Fock setting, the order 1 of the perturbation coincides with the 'central limit scaling' studied by Zelditch-Zhou for Toeplitz operators.

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Section: Statement Of Results In the Bargmann-fock Settingmentioning

confidence: 99%

Section: Conjugation To the Bargmann-fock Modelmentioning

confidence: 99%

Section: Conjugation To the Bargmann-fock Modelmentioning

confidence: 99%

Section: Conjugation To the Bargmann-fock Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Refined Weyl Law for Homogeneous Perturbations of the Harmonic Oscillator

Doll

Gannot²,

Wunsch³

2018

Commun. Math. Phys.

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Pointwise Weyl Law for Partial Bergman Kernels

Zelditch

Zhou²

2018

Algebraic and Analytic Microlocal Analysis

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This article is a continuation of a series by the authors on partial Bergman kernels and their asymptotic expasions. We prove a 2-term pointwise Weyl law for semi-classical spectral projections onto sums of eigenspaces of spectral width = k −1 of Toeplitz quantizationsĤ k of Hamiltonians on powers L k of a positive Hermitian holomorphic line bundle L → M over a Kähler manifold. The first result is a complete asymptotic expansion for smoothed spectral projections in terms of periodic orbit data. When the orbit is 'strongly hyperbolic' the leading coefficient defines a uniformly continuous measure on R and a semi-classical Tauberian theorem implies the 2-term expansion. As in previous works in the series, we use scaling asymptotics of the Boutet-de-Monvel-Sjostrand parametrix and Taylor expansions to reduce the proof to the Bargmann-Fock case.This article is part of a series [ZZ16, ZZ17] devoted to partial Bergman kernels on polarized (mainly compact) Kähler manifolds (L, h) → (M m , ω, J), i.e. Kähler manifolds of (complex) dimension m equipped with a Hermitian holomorphic line bundle whose curvature form is ω h = ω. Partial Bergman kernelsThe asymptotics of µ z,1,E k (f ) depend on whether or not z ∈ M is a periodic point for g t .Definition 0.1. Define periodic points of g t , as follows:For z ∈ P E , let T z denote the minimal period T > 0 of z.It may occur that z ∈ P E but the orbit g t h (x) with π(x) = z is not periodic, where g t h is the flow generated by the horizontal lift ξ h H of the Hamiltonian vector field ξ H . This is due to holonomy effects: parallel translation of sections of L k around the closed curve t → g t (z) may have non-trivial holonomy. We denote the holonomy by e inθ h z := the unique element e iθ ∈ S 1 : g nTz h x = r θ x.Let z ∈ P E , T = nT z be a period for n ∈ Z. Then Dg T z induces linear symplectic mapWhen working in the Kähler context it is better to conjugate to the complexifications,We denote the projection to the 'holomorphic component' byThe spaces T 1,0 M, T 0,1 M are paired complex Lagrangian subspaces.Relative to a symplectic basis {e j , Je k } of T z M in which J assumes the standard form J 0 , the matrix of Dg nTz has the form, Dg nTz z

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Edge Behavior of Higher Complex-Dimensional Determinantal Point Processes

Leslie

2023

Ann. Henri Poincaré

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As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of complex random normal matrix models that one finds a complementary error function behavior at the boundary (also called edge) of the droplet as the matrix size increases. Such behavior is seen both in the density of the eigenvalues and the correlation kernel, where the Faddeeva plasma kernel emerges. These results are neatly expressed with the help of the outward unit normal vector on the edge. We prove that such universal behaviors transcend this class of random normal matrices, being also valid in a specific “elliptic” class of determinantal point processes defined on $$\mathbb C^d$$ C d , which are higher dimensional generalizations of the determinantal point processes describing the eigenvalues of the complex Ginibre ensemble and the complex elliptic Ginibre ensemble. These models describe a system of particles in $$\mathbb C^d$$ C d with mutual repulsion, that are confined to the origin by an external field $${\mathscr {V}}(z) = |z|^2 - \tau {\text {Re}}(z_1^2+\ldots +z_d^2)$$ V ( z ) = | z | 2 - τ Re ( z 1 2 + … + z d 2 ) , where $$0\le \tau <1$$ 0 ≤ τ < 1 . Their average density of particles converges to a uniform law on a 2d-dimensional ellipsoidal region. It is on the hyperellipsoid bounding this region that we find a complementary error function behavior and the Faddeeva plasma kernel. To the best of our knowledge, this is the first instance of the Faddeeva plasma kernel emerging in a higher dimensional model. The results provide evidence for a possible edge universality theorem for determinantal point processes on $$\mathbb C^d$$ C d .

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Central limit theorem for spectral partial Bergman kernels

Cited by 24 publications

References 26 publications

Refined Weyl Law for Homogeneous Perturbations of the Harmonic Oscillator

Refined Weyl Law for Homogeneous Perturbations of the Harmonic Oscillator

Pointwise Weyl Law for Partial Bergman Kernels

Edge Behavior of Higher Complex-Dimensional Determinantal Point Processes

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