| Tomoyuki Shirai | Last modified date：2023.10.02 |
Professor /
Division of Fundamental mathematics /
Institute of Mathematics for Industry
Papers
| 1. | Tomoyuki Shirai, Persistent homology and its application to chainlets in the Bitcoin graph, JPS Conference Proceedings of Blockchain Kaigi 2022 (BCK2022) , https://doi.org/10.7566/JPSCP.40.011, 40, 011001, 1-14, 2023.01, We give a brief explanation of homology and persistent homology intuitively by using matrix representations of boundary operators and introduce a result of the law of large numbers for persistence diagrams of a stationary ergodic point process. We recall the notion of Bitcoin graphs and chainlets and show an example of how to compute persistence diagrams for chainlet matrices by viewing them as point clouds.. |
| 2. | Makoto Katori and Tomoyuki Shirai, Local universality of determinantal point processes on Riemannian manifolds, Proc. Japan Acad. Ser. A Math. Sci. 98 (2022) 95-100. oepn access, https://doi.org/10.3792/pjaa.98.018, 98, 95-100, 2022.10. |
| 3. | Kohei Noda and Tomoyuki Shirai, Expected number of zeros of random power series with finitely dependent Gaussian coefficients, Journal of Theoretical Probability, https://doi.org/10.1007/s10959-022-01203-y, 36, 1534-1554, 2022.11, We are concerned with zeros of random power series with coefficients being a stationary, centered, complex Gaussian process. We show that the expected number of zeros in every smooth domain in the disk of convergence is less than that of the hyperbolic GAF with i.i.d. coefficients. When coefficients are finitely dependent, i.e., the spectral density is a trigonometric polynomial, we derive precise asymptotics of the expected number of zeros inside the disk of radius $r$ centered at the origin as $r$ tends to the radius of convergence, in the proof of which we clarify that the negative contribution to the number of zeros stems from the zeros of the spectral density.. |
| 4. | Tomoyuki Shirai and Kiyotaka Suzaki, Limit theorem for persistence diagrams of random filtered complexex built over marked point processes, Modern Stochastics: Theory and Applications, DOI 10.15559/22-VMSTA214, 1-18, 2022.09, Random filtered complexes built over marked point processes on Euclidean spaces are considered. Examples of these filtered complexes include a filtration of Cˇech complexes of a family of sets with various sizes, growths, and shapes. The law of large numbers for persistence diagrams is established as the size of the convex window observing a marked point process tends to infinity.. |
| 5. | 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:𝑞0 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→∞).. |
| 6. | 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.. |
| 7. | 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. |
| 8. | Takato Matsui, Makoto Katori and Tomoyuki Shirai, Local number variances and hyperuniformity of the Heisenberg family of determinantal point processes , Journal of Physics A: Mathematical and Theoretical , https://doi.org/10.1088/1751-8121/abecaa, 54, 165201 (22pp), 2021.04. |
| 9. | Yasuaki Hiraoka, Hiroyuki Ochiai, and Tomoyuki Shirai, Zeta functions of periodic cubical lattices and cyclotomic-like polynomials, Advanced Studies in Pure Mathematics, 84, 93-121, 2020.05. |
| 10. | Makoto Katori and Tomoyuki Shirai, Scaling limit for determinantal point processes on spheres, RIMS Kôkyûroku Bessatsu, B79, 123-138, 2020.04, [URL]. |
| 11. | Hidekazu Yoshioka, Tomoyuki Shirai, Daisuke Tagami, A mixed optimal control approach for upstream fish Migration, Journal of Sustainable Development of Energy, Water and Environment Systems, 10.13044/j.sdewes.d6.0221, 7, 1, 101-121, 2019.03, This paper proposes a simple mathematical model forupstream fish migration along rivers. The model describes the fish migration along a river based on a mixed optimal control approach having swimming velocity, school size, and stopping time of migration as control variables. The optimization problem reduces to a variational inequality. Its explicit “viscosity” solution is presented with the dependence of the fish migration on river environment. To prove uniqueness of the solution to the variational inequality requires a constructive argument not based on the conventional theorems. A novel finite difference scheme for solving the variational inequality is also proposed with its convergence results. An application example of the model discusses the upstream migration of Plecoglossus altivelis (Ayu) in Japan, which evaluates the dependence of the fish migration on the habitat quality and provides recommendations for managing river environment. This is an interdisciplinary research between environmental and mathematical fields.. |
| 12. | Yuji Hamana, Hiroyuki Matsumoto, Tomoyuki Shirai, On the zeros of the MacDonald functions, Opuscula Mathematica, 10.7494/OpMath.2019.39.3.361, 39, 3, 2019.01, We are concerned with the zeros of the Macdonald functions or the modified Bessel functions of the second kind with real index. By using the explicit expressions for the algebraic equations satisfied by the zeros, we describe the behavior of the zeros when the index moves. Results by numerical computations are also presented.. |
| 13. | 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.. |
| 14. | Fei Jiang, Takeshi Tsuji, Tomoyuki Shirai, Pore Geometry Characterization by Persistent Homology Theory, Water Resources Research, 10.1029/2017WR021864, 54, 6, 4150-4163, 2018.06, Rock pore geometry has heterogeneous characteristics and is scale dependent. This feature in a geological formation differs significantly from artificial materials and makes it difficult to predict hydrologic and elastic properties. To characterize pore heterogeneity, we propose an evaluation method that exploits the recently developed persistent homology theory. In the proposed method, complex pore geometry is first represented as sphere cloud data using a pore-network extraction method. Then, a persistence diagram (PD) is calculated from the point cloud, which represents the spatial distribution of pore bodies. A new parameter (distance index H) derived from the PD is proposed to characterize the degree of rock heterogeneity. Low H value indicates high heterogeneity. A new empirical equation using this index H is proposed to predict the effective elastic modulus of porous media. The results indicate that the proposed PD analysis is very efficient for extracting topological feature of pore geometry.. |
| 15. | Takayuki Osogami, Rudy Raymond, Akshay Goel, Tomoyuki Shirai, Takanori Maehara, Dynamic determinantal point processes, Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18), 3868-3875, 2018.02, 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(M^2 K) time to O(M K^2) 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.. |
| 16. | 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}).. |
| 17. | Naoto Miyoshi, Tomoyuki Shirai, Tail asymptotics of signal-to-interference ratio distribution in spatial cellular network models, Probability and Mathematical Statistics, 10.19195/0208-4147.37.2.12, 37, 2, 431-453, 2017.01, We consider a spatial stochastic model of wireless cellular networks, where the base stations (BSs) are deployed according to a simple and stationary point process on Rd, d ≥ 2. In this model, we investigate tail asymptotics of the distribution of signal-to-interference ratio (SIR), which is a key quantity in wireless communications. In the case where the pathloss function representing signal attenuation is unbounded at the origin, we derive the exact tail asymptotics of the SIR distribution under an appropriate sufficient condition. While we show that widely-used models based on a Poisson point process and on a determinantal point process meet the sufficient condition, we also give a counterexample violating it. In the case of bounded path-loss functions, we derive a logarithmically asymptotic upper bound on the SIR tail distribution for the Poisson-based and α-Ginibrebased models. A logarithmically asymptotic lower bound with the same order as the upper bound is also obtained for the Poisson-based model.. |
| 18. | 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.. |
| 19. | Tomoyuki Shirai, Evgeny Verbitskiy, Solvable and algebraic systems on infinite ladder, Indagationes Mathematicae, 10.1016/j.indag.2016.02.003, 27, 5, 1162-1183, 2016.12, We consider two solvable models with equal entropy on the infinite ladder graph Z×{1,2}: the uniform spanning forest (USF), the abelian sandpile (ASM). We show that the symbolic models (abelian sandpile and spanning forest) are equal entropy symbolic covers of a certain algebraic dynamical system. In the past results of this nature have been established for sandpile models on lattices Zd. But we present a first example in case of spanning trees.. |
| 20. | Naoto Miyoshi, Tomoyuki Shirai, Spatial modeling and analysis of cellular networks using the ginibre point process A tutorial, IEICE Transactions on Communications, 10.1587/transcom.2016NEI0001, E99B, 11, 2247-2255, 2016.11, Spatial stochastic models have been much used for perfor mance analysis of wireless communication networks. This is due to th fact that the performance of wireless networks depends on the spatial con figuration of wireless nodes and the irregularity of node locations in a rea wireless network can be captured by a spatial point process. Most work on such spatial stochastic models of wireless networks have adopted homo geneous Poisson point processes as the models of wireless node locations While this adoption makes the models analytically tractable, it assume that the wireless nodes are located independently of each other and thei spatial correlation is ignored. Recently, the authors have proposed to adop the Ginibre point process-one of the determinantal point processes-a the deployment models of base stations (BSS) in cellular networks. Th determinantal point processes constitute a class of repulsive point processe and have been attracting attention due to their mathematically interestin properties and efficient simulation methods. In this tutorial, we provide brief guide to the Ginibre point process and its variant, -Ginibre poin process, as the models of BS deployments in cellular networks and sho some existing results on the performance analysis of cellular network mod els with -Ginibre deployed BSS. The authors hope the readers to use suc point processes as a tool for analyzing various problems arising in futur cellular networks.. |
| 21. | 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. |
| 22. | 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. |
| 23. | 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. |
| 24. | Naoto Miyoshi, Tomoyuki Shirai, A cellular network model with Ginibre configured base stations, Advances in Applied Probability, 10.1239/aap/1409319562, 46, 3, 832-845, 2014.09, Stochastic geometry models for wireless communication networks have recently attracted much attention. This is because the performance of such networks critically depends on the spatial configuration of wireless nodes and the irregularity of the node configuration in a real network can be captured by a spatial point process. However, most analysis of such stochastic geometry models for wireless networks assumes, owing to its tractability, that the wireless nodes are deployed according to homogeneous Poisson point processes. This means that the wireless nodes are located independently of each other and their spatial correlation is ignored. In this work we propose a stochastic geometry model of cellular networks such that the wireless base stations are deployed according to the Ginibre point process. The Ginibre point process is one of the determinantal point processes and accounts for the repulsion between the base stations. For the proposed model, we derive a computable representation for the coverage probability-the probability that the signal-to-interference-plus-noise ratio (SINR) for a mobile user achieves a target threshold. To capture its qualitative property, we further investigate the asymptotics of the coverage probability as the SINR threshold becomes large in a special case. We also present the results of some numerical experiments.. |
| 25. | Sho Matsumoto, Tomoyuki Shirai, Correlation functions for zeros of a Gaussian power series and Pfaffians, Electronic Journal of Probability, 10.1214/EJP.v18-2545, 18, 2013.04, We show that the zeros of the random power series with i.i.d. real Gaussian coefficients form a Pfaffian point process. We further show that the product moments for absolute values and signatures of the power series can also be expressed by Pfaffians.. |
| 26. | Takayuki Osogami, Tomoyuki Shirai, Hayato Waki, Remarks on positivity of α-determinants via SDP relaxation, Journal of Math-for-Industry, 5, A, 1-10, 2013.04. |
| 27. | Tomoyuki Shirai, Fumio Hiroshima, Itaru Sasaki, Akito Suzuki, Note on the spectrum of discrete Schrodinger operators, Journal of Math-for-Industry, 2012B4, 105-108, 2012.09. |
| 28. | Tomoyuki Shirai, Limit theorem for random analytic functions and their zeros, RIMS Kôkyûroku Bessatsu, to appear, 2012.07. |
| 29. | 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. |
| 30. | Tomoyuki Shirai, A remark on monotonicity for the Glauber dynamics on finite graphs, Proceedings of the Japan Academy Series A: Mathematical Sciences, 10.3792/pjaa.86.33, 86, 2, 33-37, 2010.02, We show that under the heat-bath Glauber dynamics for the ferromagnetic Ising model on a finite graph, the single spin expectation at a fixed time starting at the all-up configuration is not necessarily an increasing function of coupling constants.. |
| 31. | Takuya Ohwa, Tomoyuki Shirai, Joint distribution of the cover time and the last visited point of finite Markov chains, Kyushu Journal of Mathematics, 10.2206/kyushujm.62.281, 62, 1, 281-292, 2008.06, We consider a Markov chain on a finite state space and obtain an expression of the joint distribution of the cover time and the last point visited by the Markov chain. As a corollary, we obtain the spectral representation of the distribution of the cover time.. |
| 32. | Hirofumi Osada, Tomoyuki Shirai, Variance of the linear statistics of the Ginibre random point field, RIMS Kôkyûroku Bessatsu Proceedings of RIMS Workshop on Stochastic Analysis and Applications, B6, 193-200, 2008.06. |
| 33. | Tomoyuki Shirai, Remarks on the positivity of α-determinants, Kyushu Journal of Mathematics, 10.2206/kyushujm.61.169, 61, 1, 169-189, 2007.06, We give a probabilistic expression of the α-determinant by using the Wishart distribution on the cone of positive definite matrices. As a corollary, we give a partial answer to a positivity problem for the α-determinant which is equivalent to the existence problem of certain multivariate probability distributions. We also give a concrete example to support our conjecture for the positivity.. |
| 34. | Yusuke Higuchi, Tomoyuki Shirai, Non-bipartiteness of graphs and the upper bounds of Dirichlet forms, Potential Analysis, 10.1007/s11118-006-9027-z, 25, 3, 259-268, 2006.11, The sum of the lower bound and the upper one of the spectrum of our discrete Laplacian is less than or equal to 2. The equality holds if a graph is bipartite while the converse does not hold for general infinite graphs. In this paper, we give an estimate of the upper bounds of Dirichlet forms and using this estimate together with an h-transform, we show that the sum is strictly less than 2 for a certain class of infinite graphs.. |
| 35. | Tomoyuki Shirai, Large deviations for the fermion point process associated with the exponential kernel, Journal of Statistical Physics, 10.1007/s10955-006-9026-x, 123, 3, 615-629, 2006.05, For the fermion point process on the whole complex plane associated with the exponential kernel ezw̄, we show the central limit theorem for the random variable ξ(D r , the number of points inside the ball D r of radius r, as r → ∞ and we establish the large deviation principle for the random variables {r -2ξ (D r ), r > 0}.. |
| 36. | Tomoyuki Shirai, Yoichiro Takahashi, Random point fields associted with fermion, boson and other statistics, Adv. Stud. Pure Math., Vol.39, 345--354, 2004.01. |
| 37. | Tomoyuki Shirai, Yu. Higuchi, Isoperimetric constants of (d,f)-regular planar graphs, Interdisp. Inform. Sci., Vol. 9, 221--228., 2003.01. |
| 38. | Tomoyuki Shirai, Yu.Higuchi, Some spectral and geometric properties for infinite graphs, Contemporary Mathematics 347, Vol. 347, 29--56, 2004.01. |
| 39. | 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.. |
| 40. | 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.. |
| 41. | Tomoyuki Shirai, Long time behavior of the transition probability of a random walk with drift on an abelian covering graph, Tohoku Mathematical Journal, 10.2748/tmj/1113246940, 55, 2, 255-269, 2003.01, For a certain class of reversible random walks possibly with drift on an abelian covering graph of a finite graph, using the technique of twisted transition operator, we obtain the asymptotic behavior of the n-step transition probability pn(x, y) as n → − give an expression of the constant which appears in the asymptotics.. |
| 42. | Tomoyuki Shirai, Hyun Jae Yoo, Glauber dynamics for fermion point processes, Nagoya Mathematical Journal, 10.1017/S0027763000008412, 168, 139-166, 2002.01, 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.. |
| 43. | Yusuke Higuchi, Tomoyuki Shirai, Weak Bloch property for discrete magnetic Schrodinger operators, Nagoya Mathematical Journal, 10.1017/S0027763000022157, 161, 127-154, 2001.01, For a magnetic Schrödinger operator on a graph, which is a generalization of classical Harper operator, we study some spectral properties: the Bloch property and the behaviour of the bottom of the spectrum with respect to magnetic fields. We also show some examples which have interesting properties.. |
| 44. | Tomoyuki Shirai, Asymptotic behavior of the transition probability of a simple random walk on a line graph, Journal of the Mathematical Society of Japan, 10.2969/jmsj/05210099, 52, 1, 99-108, 2000.01, For simple random walks (Pn G) on a homogeneous graph G and (Pn L(G)) on its line graph L(G), we obtain the relationship between the asymptotic behavior of the n-step transition probability Pn G(x, x) and that of Pn L(G)(x, x) as n→∞.. |
| 45. | Tomoyuki Shirai, The spectrum of infinite regular line graphs, Transactions of the American Mathematical Society, 352, 1, 115-132, 2000.01, Let G be an infinite d-regular graph and L(G) its line graph. We consider discrete Laplacians on G and L(G), and show the exact relation between the spectrum of -Δc and that of -ΔL(G), Our method is also applicable to (d1,d2)-seiniregular graphs, subdivision graphs and para-line graphs.. |
| 46. | Tomoyuki Shirai, Yusuke Higuchi, A remark on the spectrum of magnetic Laplacian on a graph, Yokohama Mathematical Journal, Vol.47, 129-142., 1999.01. |
| 47. | Yusuke Higuchi, Tomoyuki Shirai, The Spectrum of Magnetic Schrödinger Operators on a Graph with Periodic Structure, Journal of Functional Analysis, 10.1006/jfan.1999.3478, 169, 2, 456-480, 1999.12, For discrete magnetic Schrödinger operators on covering graphs of a finite graph, we investigate two spectral properties: (1) the relationship between the spectrum of the operator on the covering graph and that on a finite graph, (2) the analyticity of the bottom of the spectrum with respect to magnetic flow. Also we compute the second derivative of the bottom of the spectrum and represent it in terms of geometry of a graph.. |
| 48. | Motoko Kotani, Tomoyuki Shirai, Toshikazu Sunada, Asymptotic Behavior of the Transition Probability of a Random Walk on an Infinite Graph, Journal of Functional Analysis, 10.1006/jfan.1998.3322, 159, 2, 664-689, 1998.11, Ideas cultivated in spectral geometry are applied to obtain an asymptotic property of a reversible random walk on an infinite graph satisfying a certain periodic condition. In the course of our argument, we employ perturbation theory for the maximal eigenvalues of twisted transition operator. As a result, an asymptotic of the probabilityp(n,x,y) that a particle starting atxreachesyat timenasngoes to infinity is established.. |
| 49. | Tomoyuki Shirai, A factorization of determinant related to some random matrices, Journal of Statistical Physics, 90, 5-6, 1449-1459, 1998.03, We consider the expectation of the determinant del(λ - X)-1 for Im λ > 0 associated with some random NxN matrices and factorize it into N Stieltjes transforms of probability measures. Moreover, using this factorization, we investigate the limiting behavior of the logarithm of the quantity as N → ∞.. |
| 50. | Tomoyuki Shirai, A trace formula for discrete Schrödinger operators, Publications of the Research Institute for Mathematical Sciences, 10.2977/prims/1195144826, 34, 1, 27-41, 1998.01, We discuss two types of trace formula which arise from the inverse spectral problem for discrete Schrödinger operators as L = -Δ + V (x) where V is a bounded potential. One is the relationship between a potential and spectral data, and another is the one between the green function of L and periodic orbits of a state space.. |
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