| 1. |
Makoto Katori and Tomoyuki Shirai, Zeros of the i.i.d. Gaussian Laurent Series on an Annulus: Weighted Szegő Kernels and Permanental-Determinantal Point Processes, Communications in Mathematical Physics, https://doi.org/10.1007/s00220-022-04365-2, 392, 1099-1151, 2022.06, On an annulus 𝔸𝑞:={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 𝔸𝑞 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 𝔸𝑞 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↔𝑞2/𝑟. When 𝑟=𝑞 they are invariant under conformal transformations which preserve 𝔸𝑞. 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→∞).. |
| 2. |
Subhroshekhar Ghosh, Naoto Miyoshi and Tomoyuki Shirai, Disordered complex networks: energy optimal lattices and persistent homology, IEEE Transactions on Information Theory, doi: 10.1109/TIT.2022.3163604, 2022.04, Disordered complex networks are of fundamental interest in statistical physics, and they have attracted recent interest as stochastic models for information transmission over wireless networks. While mathematically tractable, a network based on the regulation Poisson point process model offers challenges vis-a-vis network efficiency. Strongly correlated alternatives, such as networks based on random matrix spectra (the Ginibre network), on the other hand offer formidable challenges in terms of tractability and robustness issues. In this work, we demonstrate that network models based on random perturbations of Euclidean lattices interpolate between Poisson and rigidly structured networks, and allow us to achieve the best of both worlds: significantly improve upon the Poisson model in terms of network efficacy measured by the Signal to Interference plus Noise Ratio (abbrv. SINR) and the related concept of coverage probabilities, at the same time retaining a considerable measure of mathematical and computational simplicity and robustness to erasure and noise. We investigate the optimal choice of the base lattice in this model, connecting it to the celebrated problem optimality of Euclidean lattices with respect to the Epstein Zeta function, which is in turn related to notions of lattice energy. This leads us to the choice of the triangular lattice in 2D and face centered cubic lattice in 3D, whose Gaussian perturbations we consider. We provide theoretical analysis and empirical investigations to demonstrate that the coverage probability decreases with increasing strength of perturbation, eventually converging to that of the Poisson network. In the regime of low disorder, our studies suggest an approximate statistical behaviour of the coverage function near a base station as a log-normal distribution with parameters depending on the Epstein Zeta function of the lattice, and related approximate dependencies for a power-law constant that governs the network coverage probability at large thresholds. In 2D, we determine the disorder strength at which the perturbed triangular lattice (abbrv. PTL) and the Ginibre networks are the closest measured by comparing their network topologies via a comparison of their Persistence Diagrams in the total variation as well as the symmetrized nearest neighbour distances. We demonstrate that, at this very same disorder, the PTL and the Ginibre networks exhibit very similar coverage probability distributions, with the PTL performing at least as well as the Ginibre. Thus, the PTL network at this disorder strength can be taken to be an effective substitute for the Ginibre network model, while at the same time offering the advantages of greater tractability both from theoretical and empirical perspectives.. |
| 3. |
Makoto Katori and Tomoyuki Shirai, Partial Isometry, Duality, and Determinantal Point Processes, Random Matrices: Theory and Applications, https://dx.doi.org/10.1142/S2010326322500253, 2250025 (70 pages), 2021.10. |
| 4. |
Yasuaki Hiraoka, Tomoyuki Shirai, Khanh Duy Trinh, Limit theorems for persistence diagrams, Annals of Applied Probability, 10.1214/17-AAP1371, 28, 5, 2740-2780, 2018.10, The persistent homology of a stationary point process on RN 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 support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.. |
| 5. |
Yasuaki Hiraoka, Tomoyuki Shirai, Minimum spanning acycle and lifetime of persistent homology in the Linial–Meshulam process, Random Structures and Algorithms, 10.1002/rsa.20718, 51, 2, 315-340, 2017.09, This paper studies a higher dimensional generalization of Frieze's ζ(3) -limit theorem on the d-Linial–Meshulam 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 Θ (n^{d-1}).. |
| 6. |
Alexander Igorevich Bufetov, Tomoyuki Shirai, Quasi-symmetries and rigidity for determinantal point processes associated with de Branges spaces, Proceedings of the Japan Academy Series A: Mathematical Sciences, 10.3792/pjaa.93.1, 93, 1, 1-5, 2017.01, 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.. |
| 7. |
Tomoyuki Shirai, Trinh Khanh Duy, The mean spectral measures of random Jacobi matrices related Gaussian beta ensembles, Electoric Communications of Probability, 10.1214/ECP.v20-4252, 20, 68, 1-13, 2015.10. |
| 8. |
Tomoyuki Shirai, Hirofumi Osada, Absolute continuity and singularity of Palm measures of the Ginibre point process, Probability Theory and Related Fields, 10.1007/s00440-015-0644-6, 20, 68, 725-770, 2015.07. |
| 9. |
Tomoyuki Shirai, Ginibre-type point processes and their asymptotic behavior, JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, 10.2969/jmsj/06720763, 67, 2, 763-787, 2015.04. |
| 10. |
Tomoyuki Shirai, Limit theorem for random analytic functions and their zeros, RIMS Kôkyûroku Bessatsu, to appear, 2012.07. |
| 11. |
Takuya Ohwa, Yusuke Higuchi and Tomoyuki Shirai, Exact computation for the cover times of certain classes of trees, Journal of Math-for-Industry, 2, A, 93-98, 2010.04. |
| 12. |
Tomoyuki Shirai, Yoichiro Takahashi, Random point fields associated with certain Fredholm determinants I
Fermion, poisson and boson point processes, Journal of Functional Analysis, 10.1016/S0022-1236(03)00171-X, 205, 2, 414-463, 2003.12, We introduce certain classes of random point fields, including fermion and boson point processes, which are associated with Fredholm determinants of certain integral operators and study some of their basic properties: limit theorems, correlation functions, Palm measures etc. Also we propose a conjecture on an α-analogue of the determinant and permanent.. |
| 13. |
Tomoyuki Shirai, Yoichiro Takahashi, Random point fields associated with certain Fredholm determinants II
Fermion shifts and their ergodic and Gibbs properties, Annals of Probability, 10.1214/aop/1055425789, 31, 3, 1533-1564, 2003.07, We construct and study a family of probability measures on the configuration space over countable discrete space associated with nonnegative definite symmetric operators via determinants. Under a mild condition they turn out unique Gibbs measures. Also some ergodic properties, including the entropy positivity, are discussed in the lattice case.. |
