Updated on 2024/04/18

写真a

 
TRINH, Khanh Duy
 
Affiliation
Faculty of Science and Engineering, Global Center for Science and Engineering
Job title
Associate Professor(without tenure)

Research Experience

  • 2019.04
    -
    Now

    Waseda University   Global Center for Science and Engineering   Associate Professor (without tenure)

  • 2017.12
    -
    2019.03

    Research Alliance Center for Mathematical Sciences, Tohoku University, Japan   Associate Professor

  • 2013.10
    -
    2017.11

    Institute of Mathematics for Industry, Kyushu University, Japan   Assistant Professor

  • 2012.07
    -
    2013.09

    Department of Mathematics, Graduate School of Science, Osaka University, Japan   Research Fellow of the Japan Society for the Promotion of Science

Research Areas

  • Basic analysis   Probability theory and applied probability
 

Papers

  • Random connection models in the thermodynamic regime: central limit theorems for add-one cost stabilizing functionals

    Van Hao Can, Khanh Duy Trinh

    Electronic Journal of Probability   27  2022

     View Summary

    The paper deals with a random connection model, a random graph whose vertices are given by a homogeneous Poisson point process on Rd, and edges are independently drawn with probability depending on the locations of the two end points. We establish central limit theorems (CLT) for general functionals on this graph under minimal assumptions that are a combination of the weak stabilization for the add-one cost and a (2 + δ)-moment condition. As a consequence, CLTs for isomorphic subgraph counts, isomorphic component counts, the number of connected components are then derived. In addition, CLTs for Betti numbers and the size of the largest component are also proved for the first time.

    DOI

    Scopus

    1
    Citation
    (Scopus)
  • Beta Jacobi Ensembles and Associated Jacobi Polynomials

    Hoang Dung Trinh, Khanh Duy Trinh

    Journal of Statistical Physics   185 ( 1 )  2021.10

     View Summary

    Beta ensembles on the real line with three classical weights (Gaussian, Laguerre and Jacobi) are now realized as the eigenvalues of certain tridiagonal random matrices. The paper deals with beta Jacobi ensembles, the type with the Jacobi weight. Making use of the random matrix model, we show that in the regime where βN→ const∈ [0 , ∞) , with N the system size, the empirical distribution of the eigenvalues converges weakly to a limiting measure which belongs to a new class of probability measures of associated Jacobi polynomials. This is analogous to the existing results for the other two classical weights. We also study the limiting behavior of the empirical measure process of beta Jacobi processes in the same regime and obtain a dynamical version of the above.

    DOI

    Scopus

    7
    Citation
    (Scopus)
  • Beta Laguerre processes in a high temperature regime

    Hoang Dung Trinh, Khanh Duy Trinh

    Stochastic Processes and their Applications   136   192 - 205  2021.06

     View Summary

    Beta Laguerre processes which are generalizations of the eigenvalue process of Wishart/Laguerre processes can be defined as the squares of radial Dunkl processes of type B. In this paper, we study the limiting behavior of their empirical measure processes. By the moment method, we show the convergence to a limit in a high temperature regime, a regime where βN→const∈(0,∞), where β is the inverse temperature parameter and N is the system size. This is a dynamic version of a recent result on the convergence of the empirical measures of beta Laguerre ensembles in the same regime.

    DOI

    Scopus

    3
    Citation
    (Scopus)
  • Beta Laguerre ensembles in global regime

    Hoang Dung Trinh, Khanh Duy Trinh

    Osaka J. Math.   58 ( 2 ) 435 - 450  2021.04  [Refereed]

     View Summary

    Beta Laguerre ensembles, generalizations of Wishart and Laguerre ensembles, can be realized as eigenvalues of certain random tridiagonal matrices. Analogous to the Wishart (β = 1) case and the Laguerre (β = 2) case, for fixed β, it is known that the empirical distribution of the eigenvalues of the ensembles converges weakly to Marchenko–Pastur distributions, almost surely. The paper restudies the limiting behavior of the empirical distribution but in regimes where the parameter β is allowed to vary as a function of the matrix size N. We show that the above Marchenko–Pastur law holds as long as βN →∞.WhenβN → 2c ∈ (0, ∞), the limiting measure is related to associated Laguerre orthogonal polynomials. Gaussian fluctuations around the limit are also studied.

  • Poisson Statistics for Beta Ensembles on the Real Line at High Temperature

    Fumihiko Nakano, Khanh Duy Trinh

    Journal of Statistical Physics   179 ( 2 ) 632 - 649  2020.04

     View Summary

    This paper studies beta ensembles on the real line in a high temperature regime, that is, the regime where βN→ const∈ (0 , ∞) , with N the system size and β the inverse temperature. For the global behavior, the convergence to the equilibrium measure is a consequence of a recent result on large deviation principle. This paper focuses on the local behavior and shows that the local statistics around any fixed reference energy converges weakly to a homogeneous Poisson point process.

    DOI

    Scopus

    9
    Citation
    (Scopus)
  • On central limit theorems in stochastic geometry for add-one cost stabilizing functionals

    Khanh Duy Trinh

    Electronic Communications in Probability   24 ( none ) 1 - 15  2019.12

     View Summary

    We establish central limit theorems for general functionals on binomial point processes and their Poissonized version, which extends the results of Penrose–Yukich (Ann. Appl. Probab. 11(4), 1005–1041 (2001)) to the inhomogeneous case. Here functionals are required to be strongly stabilizing for add-one cost on homogeneous Poisson point processes and to satisfy some moments conditions. As an application, a central limit theorem for Betti numbers of random geometric complexes in the subcritical regime is derived.

    DOI

  • Global Spectrum Fluctuations for Gaussian Beta Ensembles: A Martingale Approach

    Khanh Duy Trinh

    Journal of Theoretical Probability   32 ( 3 ) 1420 - 1437  2019.09

     View Summary

    The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter β is allowed to vary with the matrix size n. In particular, we show that as n→ ∞ with nβ→ ∞, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.

    DOI

    Scopus

    8
    Citation
    (Scopus)
  • Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

    Akshay Goel, Khanh Duy Trinh, Kenkichi Tsunoda

    JOURNAL OF STATISTICAL PHYSICS   174 ( 4 ) 865 - 892  2019.02  [Refereed]

     View Summary

    We establish the strong law of large numbers for Betti numbers of random ech complexes built on RN-valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on RN and the case where it is supported on a C1 compact manifold of dimension strictly less than N. The strong law is proved under very mild assumption which only requires that the common probability density function belongs to Lp spaces, for all 1p<.

    DOI

    Scopus

    12
    Citation
    (Scopus)
  • Gaussian Beta Ensembles at High Temperature: Eigenvalue Fluctuations and Bulk Statistics

    Fumihiko Nakano, Khanh Duy Trinh

    Journal of Statistical Physics   173 ( 2 ) 295 - 321  2018.10  [Refereed]

     View Summary

    We study the limiting behavior of Gaussian beta ensembles in the regime where βn= const as n→ ∞. The results are (1) Gaussian fluctuations for linear statistics of the eigenvalues, and (2) Poisson convergence of the bulk statistics. (2) is an alternative proof of the result by Benaych-Georges and Péché (J Stat Phys 161(3):633–656, 2015) with the explicit form of the intensity measure.

    DOI

    Scopus

    20
    Citation
    (Scopus)
  • On spectral measures of random Jacobi matrices

    Khanh Duy Trinh

    Osaka J. Math.   55 ( 4 ) 595 - 617  2018.10  [Refereed]

     View Summary

    The paper studies the limiting behaviour of spectral measures of random Jacobi matrices of Gaussian, Wishart and MANOVA beta ensembles. We show that the spectral measures converge weakly to a limit distribution which is the semicircle distribution, Marchenko-Pastur distributions or Kesten-McKay distributions, respectively. The Gaussian fluctuation around the limit is then investigated.

  • Limit theorems for persistence diagrams

    Yasuaki Hiraoka, Tomoyuki Shirai, Khanh Duy Trinh

    Ann. Appl. Probab.   28 ( 5 ) 2740 - 2780  2018.10  [Refereed]

     View Summary

    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.

    DOI

    Scopus

    40
    Citation
    (Scopus)
  • A remark on the convergence of Betti numbers in the thermodynamic regime

    Khanh Duy Trinh

    Pac. J. Math. Ind.   9 ( 4 ) 1 - 7  2017.03  [Refereed]

    DOI

  • Distributions of the determinants of Gaussian beta ensembles

    Khanh Duy Trinh

    RIMS Kôkyûroku   2023 ( 2023 ) 77 - 85  2017

    CiNii

  • CENTRAL LIMIT THEOREM FOR MOMENTS OF SPECTRAL MEASURES OF WIGNER MATRICES

    Trinh Khanh Duy

    OSAKA JOURNAL OF MATHEMATICS   53 ( 1 ) 141 - 160  2016.01  [Refereed]

     View Summary

    Spectral measures of Wigner matrices are investigated. The Wigner semicircle law for spectral measures is proved. Regard this as the law of large number, the central limit theorem moments of spectral measures is also derived. The proof is based on moment method and combinatorial method.

    DOI

  • The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles

    Trinh Khanh Duy, Tomoyuki Shirai

    ELECTRONIC COMMUNICATIONS IN PROBABILITY   20 ( 68 ) 1 - 13  2015.09  [Refereed]

     View Summary

    An explicit formula for the mean spectral measure of a random Jacobi matrix is derived. The matrix can be regarded as the limit of Gaussian beta ensemble (G beta E) matrices as the matrix size N tends to infinity with the constraint that N beta is a constant.

    DOI

    Scopus

    21
    Citation
    (Scopus)
  • An Introduction to Ergodic Theory

    Khanh Duy Trinh

    A Mathematical Approach to Research Problems of Science and Technology     297  2014  [Refereed]

    DOI

  • On the distribution of $k$-th power free integers, II

    Khanh Duy Trinh, Takanobu Satoshi

    Osaka J. Math.   50 ( 3 ) 687 - 713  2013.07  [Refereed]

     View Summary

    The indicator function of the set of k-th power free integers is naturally extended to a random variable X(k)({dot operator}) on (ℤ○,λ), where ℤ○ is the ring of finite integral adeles and λ is the Haar probability measure. In the previous paper, the first author noted the strong law of large numbers for {X(k)({dot operator}+n)}∞n=1, and showed the asymptotics: Eλ[(Y(k)N)2]{equivalent to}1 as N→∞, where Y(k)N(x):=N-1/2k∑Nn=1(X(k)(x+n)-1/ζ(k)). In the present paper, we prove the convergence of Eλ[(Y(k)N)2]. For this, we present a general proposition of analytic number theory, and give a proof to this.

    DOI

  • On convergence of Fourier series of Besicovitch almost periodic functions

    Trinh Khanh Duy

    LITHUANIAN MATHEMATICAL JOURNAL   53 ( 3 ) 264 - 279  2013.07  [Refereed]

     View Summary

    The paper deals with convergence of the Fourier series of q-Besicovitch almost periodic functions of the form f(t) similar to Sigma(infinity)(m=1)a(m)e(-i lambda mt)
    where {lambda m} is a Dirichlet sequence, that is, a strictly increasing sequence of nonnegative numbers tending to infinity. In particular, we show that, for 1 &lt; q &lt; a, the Fourier series of f(t) converges in norm to the function f(t) itself with usual order, which is analogous to the convergence in norm of the Fourier series of a function on [0, 2 pi]. A version of the Carleson-Hunt theorem is also investigated.

    DOI

    Scopus

    3
    Citation
    (Scopus)
  • Probabilistic aspects of Besicovitch almost periodic functions

    Khanh Duy Trinh

    PhD Thesis. Osaka University    2012.06

  • Remarks on value distributions of general Dirichlet series

    Duy Trinh Khanh

    Functions in number theory and their probabilistic aspects. RIMS Kôkyûroku Bessatsu   B34   49 - 68  2012  [Refereed]

    CiNii

  • CARLESON'S THEOREM FOR GENERAL DIRICHLET SERIES

    Trinh Khanh Duy

    ANALYTIC AND PROBABILISTIC METHODS IN NUMBER THEORY     119 - 129  2012  [Refereed]

     View Summary

    This paper deals with a general Dirichlet series of the form
    Sigma(infinity)(m=1) a(m)e(-lambda ms), s = sigma + it is an element of C,
    where a(m) is an element of C, and {lambda(m)} is a strictly increasing sequence of nonnegative numbers tending to infinity. Let A be a subgroup of Rd, the real line with discrete topology, generated by (lambda(m)). The dual group of A is a compact group (Lambda) over cap with the normalized Haar measure v. Let x(lambda) be a character on (Lambda) over cap defined by x(lambda)(x) = x(lambda)(lambda is an element of A, x is an element of (Lambda) over cap). Then {x(lambda m)} is an orthonormal system in L-2((Lambda) over cap, nu). Thus, for any square summable sequence (a(m)), that is,
    Sigma(infinity)(m=1) vertical bar a(m)vertical bar(2) &lt; infinity,
    the series
    Sigma(infinity)(m=1) a(m) chi lambda(m)
    converges in L-2((Lambda) over cap, nu). Our main result claims that this series actually converges almost everywhere (with respect to the Haar measure nu). This result is analogous to Carleson's theorem for Fourier series and has some interesting consequences.

  • On the distribution of $k$-th power free integers

    Khanh Duy Trinh

    Osaka J. Math.   48 ( 4 ) 1027 - 1045  2011.10  [Refereed]

     View Summary

    Let X (k)(n) be the indicator function of the set of k-th power free integers. In this paper, we study refinements of the density theorem, ζ being the Riemann zeta function. The method we take here is a compactification of ℤ; we extend S (k)N to a random variable on a probability space (ℤ̂, λ) in a natural way, where Ẑ is the ring of finite integral adeles and λ is the shift invariant normalized Haar measure. Then we investigate the rate of L 2-convergence of S (k)N, from which the above asymptotic result is derived.

    DOI

  • LIMIT-PERIODIC ARITHMETICAL FUNCTIONS AND THE RING OF FINITE INTEGRAL ADELES

    Trinh Khanh Duy

    LITHUANIAN MATHEMATICAL JOURNAL   51 ( 4 ) 486 - 506  2011.10  [Refereed]

     View Summary

    In this paper, we show that the ring of finite integral adeles, together with its Borel field and its normalized Haar measure, is an appropriate probability space where limit-periodic arithmetical functions can be extended to random variables. The natural extensions of additive and multiplicative functions are studied. Besides, the convergence of Fourier expansions of limit-periodic functions is proved.

    DOI

    Scopus

    4
    Citation
    (Scopus)
  • On index-2 linear implicit difference equations

    Nguyen Huu Du, Le Cong Loi, Trinh Khanh Duy, Vu Tien Viet

    LINEAR ALGEBRA AND ITS APPLICATIONS   434 ( 2 ) 394 - 414  2011.01  [Refereed]

     View Summary

    This paper deals with an index-2 notion for linear implicit difference equations (LIDEs) and with the solvability of initial value problems (IVPs) for index-2 LIDEs. Besides, the cocycle property as well as the multiplicative ergodic theorem of Oseledets type are also proved. (C) 2010 Elsevier Inc. All rights reserved.

    DOI

    Scopus

    2
    Citation
    (Scopus)
  • Degenerate cocycle with index-1 and Lyapunov exponents

    Nguyen Huu Du, Trinh Khanh Duy, Vu Tien Viet

    STOCHASTICS AND DYNAMICS   7 ( 2 ) 229 - 245  2007.06  [Refereed]

     View Summary

    This paper deals with the solvability of initial-value problem and with Lyapunov exponents for linear implicit random difference equations, i.e. the difference equations where the leading term cannot be solved. An index-1 concept for linear implicit random difference equations is introduced and a formula of solutions is given. Paper is also concerned with a version of the multiplicative theorem of Oseledets type.

    DOI DOI2

    Scopus

    8
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Research Projects

  • Spectral measures of random matrices and universality of random Jacobi matrices

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2016.04
    -
    2019.03
     

    Trinh Khanh Duy, Nakano Fumihiko

     View Summary

    Gaussian beta ensembles, a natural generalization of Gaussian orthogonal/unitary/symplectic in terms of the joint probability density functions, are now realized as eigenvalues of random symmetric tridiagonal matrices, called Jacobi matrices, with independent entries. In this research, we establish several new spectral properties of Gaussian beta ensembles such as convergence to a limit and Gaussian fluctuations around the limit of the spectral measures and of the empirical distributions. Approaches which are mainly based on the random matrix model are also applicable to a large class of random Jacobi matrices.

Misc

  • On persistent homology of random Cech complexes (Stochastic Analysis on Large Scale Interacting Systems)

    Trinh Khanh Duy

      ( 79 ) 215 - 228  2020.04

     View Summary

    The paper studies the relation between critical simplices and persistence diagrams of the Cech filtration. We show that adding a critical k-simplex into the filtration corresponds either to a point in the kth persistence diagram or a point in the (k - 1)st persistence diagram. Consequently, the number of points in persistence diagrams can be expressed in terms of the number of critical simplices. As an application, we establish some convergence results related to persistence diagrams of the Cech filtrations built over binomial point processes.

    CiNii

 

Syllabus

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Research Institute

  • 2022
    -
    2024

    Waseda Research Institute for Science and Engineering   Concurrent Researcher