Papers by Tomoyuki Shirai
arXiv (Cornell University), Dec 26, 2016
The persistent homology of a stationary point process on R N is studied in this paper. As a gener... more The persistent homology of a stationary point process on R N is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the limiting persistence diagram to have the full support. We also discuss a central limit theorem for persistent Betti numbers.
arXiv (Cornell University), Mar 14, 2022
The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the ... more The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the eigenvalues lie on the unit circle in the complex plane and they can be regarded as a determinantal point process on S 1 . It is also known that the scaled point processes converge weakly to the determinantal point process associated with the so-called sine kernel as the size of matrices tends to ∞. We extend this result to the case of high-dimensional spheres and show that the scaling limit processes are determinantal point processes associated with the kernels expressed by the Bessel functions of the first kind.
arXiv (Cornell University), Mar 14, 2022
We consider the Laplace-Beltrami operator ∆ g on a smooth, compact Riemannian manifold (M, g) and... more We consider the Laplace-Beltrami operator ∆ g on a smooth, compact Riemannian manifold (M, g) and the determinantal point process X λ on M associated with the spectral projection of -∆ g onto the subspace corresponding to the eigenvalues up to λ 2 . We show that the pull-back of X λ by the exponential map exp p : T * p M → M under a suitable scaling converges weakly to the universal determinantal point process on T * p M as λ → ∞.
Proceedings of the Japan Academy. Series A, Mathematical sciences, 2017
In this note, we show that determinantal point processes on the real line corresponding to de Bra... more In this note, we show that determinantal point processes on the real line corresponding to de Branges spaces of entire functions are rigid in the sense of Ghosh-Peres and, under certain additional assumptions, quasi-invariant under the group of diffeomorphisms of the line with compact support.
Communications in Mathematical Physics, Mar 30, 2022
On an annulus A q := {z ∈ C : q < |z| < 1} with a fixed q ∈ (0, 1), we study a Gaussian analytic ... more On an annulus A q := {z ∈ C : q < |z| < 1} with a fixed q ∈ (0, 1), we study a Gaussian analytic function (GAF) and its zero set which defines a point process on A q called the zero point process of the GAF. The GAF is defined by the i.i.d. Gaussian Laurent series such that the covariance kernel parameterized by r > 0 is identified with the weighted Szegő kernel of A q with the weight parameter r studied by McCullough and Shen. The GAF and the zero point process are rotationally invariant and have a symmetry associated with the q-inversion of coordinate z ↔ q/z and the parameter change r ↔ q 2 /r . When r = q they are invariant under conformal transformations which preserve A q . Conditioning the GAF by adding zeros, new GAFs are induced such that the covariance kernels are also given by the weighted Szegő kernel of McCullough and Shen but the weight parameter r is changed depending on the added zeros. We also prove that the zero point process of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. Dependence on r of the unfolded 2-correlation function of the PDPP is studied. If we take the limit q → 0, a simpler but still non-trivial PDPP is obtained on the unit disk D. We observe that the limit PDPP indexed by r ∈ (0, ∞) can be regarded as an interpolation between the determinantal point process (DPP) on D studied by Peres and Virág (r → 0) and that DPP of Peres and Virág with a deterministic zero added at the origin (r → ∞).
arXiv (Cornell University), Mar 14, 2014
We study the eigenvalue problem for some special class of antitriangular matrices. Though the eig... more We study the eigenvalue problem for some special class of antitriangular matrices. Though the eigenvalue problem is quite classical, as far as we know, almost nothing is known about properties of eigenvalues for anti-triangular matrices. In this paper, we show that there is a nice class of anti-triangular matrices whose eigenvalues are given explicitly by their elements. Moreover, this class contains several interesting subclasses which we characterize in terms of probability measures. We also discuss the application of our main theorem to the study of interacting particle systems, which are stochastic processes studied in extensive literature.
数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu, Apr 1, 2020
The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the ... more The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the eigenvalues lie on the unit circle in the complex plane and they can be regarded as a determinantal point process on S 1 . It is also known that the scaled point processes converge weakly to the determinantal point process associated with the so-called sine kernel as the size of matrices tends to ∞. We extend this result to the case of high-dimensional spheres and show that the scaling limit processes are determinantal point processes associated with the kernels expressed by the Bessel functions of the first kind.
arXiv (Cornell University), Mar 19, 2015
This paper studies a higher dimensional generalization of Frieze's ζ(3)-limit theorem in the Erdö... more This paper studies a higher dimensional generalization of Frieze's ζ(3)-limit theorem in the Erdös-Rényi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to ζ(3) as the number of vertices goes to infinity. In this paper, we study the d-Linial-Meshulam process as a model for random simplicial complexes, where d = 1 corresponds to the Erdös-Rényi graph process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in O(n d-1 ).
Random matrices : theory and applications, Oct 22, 2021
A determinantal point process (DPP) is an ensemble of random nonnegative-integervalued Radon meas... more A determinantal point process (DPP) is an ensemble of random nonnegative-integervalued Radon measures Ξ on a space S with measure λ, whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, H , = 1, 2, which are assumed to be realized as L 2 -spaces, L 2 (S , λ ), = 1, 2, and introduce a bounded linear operator W : H 1 → H 2 and its adjoint W * : H 2 → H 1 . We show that if W is a partial isometry of locally Hilbert-Schmidt class, then we have a unique DPP (Ξ 1 , K 1 , λ 1 ) associated with W * W. In addition, if W * is also of locally Hilbert-Schmidt class, then we have a unique pair of DPPs, (Ξ , K , λ ), = 1, 2. We also give a practical framework which makes W and W * satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two-and higher-dimensional spaces S, where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter (d ∈ N) series of infinite DPPs on S = R d * Corresponding author.
Journal of Physics A, Mar 31, 2021
The bulk scaling limit of eigenvalue distribution on the complex plane C of the complex Ginibre r... more The bulk scaling limit of eigenvalue distribution on the complex plane C of the complex Ginibre random matrices provides a determinantal point process (DPP). This point process is a typical example of disordered hyperuniform system characterized by an anomalous suppression of large-scale density fluctuations. As extensions of the Ginibre DPP, we consider a family of DPPs defined on the D-dimensional complex spaces C, D ∈ N, in which the Ginibre DPP is realized when D = 1. This oneparameter family (D ∈ N) of DPPs is called the Heisenberg family, since the correlation kernels are identified with the Szegő kernels for the reduced Heisenberg group. For each D, using the modified Bessel functions, an exact and useful expression is shown for the local number variance of points included in a ball with radius R in R 2D ≃ C D . We prove that any DPP in the Heisenberg family is in the hyperuniform state of Class I, in the sense that the number variance behaves as R 2D-1 as R → ∞. Our exact results provide asymptotic expansions of the number variances in large R.
Proceedings of the AAAI Conference on Artificial Intelligence
The determinantal point process (DPP) has been receiving increasing attention in machine learning... more The determinantal point process (DPP) has been receiving increasing attention in machine learning as a generative model of subsets consisting of relevant and diverse items. Recently, there has been a significant progress in developing efficient algorithms for learning the kernel matrix that characterizes a DPP. Here, we propose a dynamic DPP, which is a DPP whose kernel can change over time, and develop efficient learning algorithms for the dynamic DPP. In the dynamic DPP, the kernel depends on the subsets selected in the past, but we assume a particular structure in the dependency to allow efficient learning. We also assume that the kernel has a low rank and exploit a recently proposed learning algorithm for the DPP with low-rank factorization, but also show that its bottleneck computation can be reduced from O(M2 K) time to O(M K2) time, where M is the number of items under consideration, and K is the rank of the kernel, which can be set smaller than M by orders of magnitude.
Journal of Physics A: Mathematical and Theoretical
The bulk scaling limit of eigenvalue distribution on the complex plane C of the complex Ginibre r... more The bulk scaling limit of eigenvalue distribution on the complex plane C of the complex Ginibre random matrices provides a determinantal point process (DPP). This point process is a typical example of disordered hyperuniform system characterized by an anomalous suppression of large-scale density fluctuations. As extensions of the Ginibre DPP, we consider a family of DPPs defined on the D-dimensional complex spaces C , D ∈ N , in which the Ginibre DPP is realized when D = 1. This one-parameter family ( D ∈ N ) of DPPs is called the Heisenberg family, since the correlation kernels are identified with the Szegő kernels for the reduced Heisenberg group. For each D, using the modified Bessel functions, an exact and useful expression is shown for the local number variance of points included in a ball with radius R in R 2 D ≃ C D . We prove that any DPP in the Heisenberg family is in the hyperuniform state of class I, in the sense that the number variance behaves as R 2D−1 as R → ∞. Our ex...
Communications in Mathematical Physics
On an annulus $${{\mathbb {A}}}_q :=\{z \in {{\mathbb {C}}}: q< |z| < 1\}$$ A q : = { z ∈ C... more On an annulus $${{\mathbb {A}}}_q :=\{z \in {{\mathbb {C}}}: q< |z| < 1\}$$ A q : = { z ∈ C : q < | z | < 1 } with a fixed $$q \in (0, 1)$$ q ∈ ( 0 , 1 ) , we study a Gaussian analytic function (GAF) and its zero set which defines a point process on $${{\mathbb {A}}}_q$$ A q called the zero point process of the GAF. The GAF is defined by the i.i.d. Gaussian Laurent series such that the covariance kernel parameterized by $$r >0$$ r > 0 is identified with the weighted Szegő kernel of $${{\mathbb {A}}}_q$$ A q with the weight parameter r studied by McCullough and Shen. The GAF and the zero point process are rotationally invariant and have a symmetry associated with the q-inversion of coordinate $$z \leftrightarrow q/z$$ z ↔ q / z and the parameter change $$r \leftrightarrow q^2/r$$ r ↔ q 2 / r . When $$r=q$$ r = q they are invariant under conformal transformations which preserve $${{\mathbb {A}}}_q$$ A q . Conditioning the GAF by adding zeros, new GAFs are induced suc...
arXiv (Cornell University), Mar 14, 2022
The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the ... more The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the eigenvalues lie on the unit circle in the complex plane and they can be regarded as a determinantal point process on S 1 . It is also known that the scaled point processes converge weakly to the determinantal point process associated with the so-called sine kernel as the size of matrices tends to ∞. We extend this result to the case of high-dimensional spheres and show that the scaling limit processes are determinantal point processes associated with the kernels expressed by the Bessel functions of the first kind.
arXiv (Cornell University), Mar 14, 2022
数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu, Apr 1, 2020
The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the ... more The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the eigenvalues lie on the unit circle in the complex plane and they can be regarded as a determinantal point process on S 1 . It is also known that the scaled point processes converge weakly to the determinantal point process associated with the so-called sine kernel as the size of matrices tends to ∞. We extend this result to the case of high-dimensional spheres and show that the scaling limit processes are determinantal point processes associated with the kernels expressed by the Bessel functions of the first kind.
Random Matrices: Theory and Applications, 2021
A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon mea... more A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures [Formula: see text] on a space S with measure [Formula: see text], whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, [Formula: see text], which are assumed to be realized as [Formula: see text]-spaces, [Formula: see text], [Formula: see text], and introduce a bounded linear operator [Formula: see text] and its adjoint [Formula: see text]. We show that if [Formula: see text] is a partial isometry of locally Hilbert–Schmidt class, then we have a unique DPP [Formula: see text] associated with [Formula: see text]. In addition, if [Formula: see text] is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, [Formula: see text], [Formula: see text]. We also give a practical framework which makes [Formula: see text] and [Formula: see text] satisfy the above co...
Random Structures & Algorithms, 2017
This paper studies a higher dimensional generalization of Frieze's ζ(3)-limit theorem in the Erdö... more This paper studies a higher dimensional generalization of Frieze's ζ(3)-limit theorem in the Erdös-Rényi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to ζ(3) as the number of vertices goes to infinity. In this paper, we study the d-Linial-Meshulam process as a model for random simplicial complexes, where d = 1 corresponds to the Erdös-Rényi graph process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in O(n d-1 ).
Probability Theory and Related Fields, 2015
We prove a dichotomy between absolute continuity and singularity of the Ginibre point process $$\... more We prove a dichotomy between absolute continuity and singularity of the Ginibre point process $$\mathsf {G}$$ G and its reduced Palm measures $$\{\mathsf {G}_{\mathbf {x}}, \mathbf {x} \in \mathbb {C}^{\ell }, \ell = 0,1,2\ldots \}$$ { G x , x ∈ C ℓ , ℓ = 0 , 1 , 2 … } , namely, reduced Palm measures $$\mathsf {G}_{\mathbf {x}}$$ G x and $$\mathsf {G}_{\mathbf {y}}$$ G y for $$\mathbf {x} \in \mathbb {C}^{\ell }$$ x ∈ C ℓ and $$\mathbf {y} \in \mathbb {C}^{n}$$ y ∈ C n are mutually absolutely continuous if and only if $$\ell = n$$ ℓ = n ; they are singular each other if and only if $$\ell \not = n$$ ℓ ≠ n . Furthermore, we give an explicit expression of the Radon–Nikodym density $$d\mathsf {G}_{\mathbf {x}}/d \mathsf {G}_{\mathbf {y}}$$ d G x / d G y for $$\mathbf {x}, \mathbf {y} \in \mathbb {C}^{\ell }$$ x , y ∈ C ℓ .
Nagoya Mathematical Journal, 2002
We construct a Glauber dynamics on {0, 1}ℛ, ℛ a discrete space, with infinite range flip rates, f... more We construct a Glauber dynamics on {0, 1}ℛ, ℛ a discrete space, with infinite range flip rates, for which a fermion point process is reversible. We also discuss the ergodicity of the corresponding Markov process and the log-Sobolev inequality.
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Papers by Tomoyuki Shirai