2024/07/18 更新

写真a

トゥリン カーン デュイ
トゥリン カーン デュイ
所属
理工学術院 国際理工学センター(理工学術院)
職名
准教授(任期付)

経歴

  • 2019年04月
    -
    継続中

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

  • 2017年12月
    -
    2019年03月

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

  • 2013年10月
    -
    2017年11月

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

  • 2012年07月
    -
    2013年09月

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

研究分野

  • 基礎解析学   Probability theory and applied probability
 

論文

  • Limit theorems for moment processes of beta Dyson’s Brownian motions and beta Laguerre processes

    Fumihiko Nakano, Hoang Dung Trinh, Khanh Duy Trinh

    Random Matrices: Theory and Applications   12 ( 03 )  2023年03月

     概要を見る

    In the regime where the parameter beta is proportional to the reciprocal of the system size, it is known that the empirical distribution of Gaussian beta ensembles (respectively, beta Laguerre ensembles) converges weakly to a probability measure of associated Hermite polynomials (respectively, associated Laguerre polynomials), almost surely. Gaussian fluctuations around the limit have been known as well. This paper aims to study a dynamical version of those results. More precisely, we study beta Dyson’s Brownian motions and beta Laguerre processes and establish law of large numbers (LLNs) and central limit theorems (CLTs) for their moment processes in the same regime.

    DOI

  • 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年

     概要を見る

    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
    被引用数
    (Scopus)
  • Beta Jacobi Ensembles and Associated Jacobi Polynomials

    Hoang Dung Trinh, Khanh Duy Trinh

    Journal of Statistical Physics   185 ( 1 )  2021年10月

     概要を見る

    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

    10
    被引用数
    (Scopus)
  • Beta Laguerre processes in a high temperature regime

    Hoang Dung Trinh, Khanh Duy Trinh

    Stochastic Processes and their Applications   136   192 - 205  2021年06月

     概要を見る

    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
    被引用数
    (Scopus)
  • Beta Laguerre ensembles in global regime

    Hoang Dung Trinh, Khanh Duy Trinh

    Osaka J. Math.   58 ( 2 ) 435 - 450  2021年04月  [査読有り]

     概要を見る

    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年04月

     概要を見る

    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

    10
    被引用数
    (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年12月

     概要を見る

    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年09月

     概要を見る

    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
    被引用数
    (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年02月  [査読有り]

     概要を見る

    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
    被引用数
    (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年10月  [査読有り]

    DOI

    Scopus

    22
    被引用数
    (Scopus)
  • On spectral measures of random Jacobi matrices

    Trinh Khanh Duy

    Osaka J. Math.   55 ( 4 ) 595 - 617  2018年10月  [査読有り]

  • Limit theorems for persistence diagrams

    Yasuaki Hiraoka, Tomoyuki Shirai, Khanh Duy Trinh

    Ann. Appl. Probab.   28 ( 5 ) 2740 - 2780  2018年10月  [査読有り]

    DOI

    Scopus

    41
    被引用数
    (Scopus)
  • A remark on the convergence of Betti numbers in the thermodynamic regime

    Khanh Duy Trinh

    Pacific Journal of Mathematics for Industry   9 ( 1 ) 1 - 7  2017年03月  [査読有り]

    DOI

  • Distributions of the determinants of Gaussian beta ensembles

    Trinh Khanh Duy

    RIMS Kôkyûroku   2023 ( 2023 ) 77 - 85  2017年

    CiNii

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

    Trinh Khanh Duy

    OSAKA JOURNAL OF MATHEMATICS   53 ( 1 ) 141 - 160  2016年01月  [査読有り]

     概要を見る

    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年09月  [査読有り]

     概要を見る

    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

    22
    被引用数
    (Scopus)
  • An Introduction to Ergodic Theory

    Khanh Duy Trinh

    A Mathematical Approach to Research Problems of Science and Technology     297  2014年  [査読有り]

    DOI

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

    Khanh Duy Trinh, Takanobu Satoshi

    Osaka J. Math.   50 ( 3 ) 687 - 713  2013年07月  [査読有り]

    DOI

  • On convergence of Fourier series of Besicovitch almost periodic functions

    Trinh Khanh Duy

    LITHUANIAN MATHEMATICAL JOURNAL   53 ( 3 ) 264 - 279  2013年07月  [査読有り]

     概要を見る

    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
    被引用数
    (Scopus)
  • Probabilistic aspects of Besicovitch almost periodic functions

    Trinh Khanh Duy

    PhD Thesis. Osaka University    2012年06月

  • 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年  [査読有り]

    CiNii

  • CARLESON'S THEOREM FOR GENERAL DIRICHLET SERIES

    Trinh Khanh Duy

    ANALYTIC AND PROBABILISTIC METHODS IN NUMBER THEORY     119 - 129  2012年  [査読有り]

     概要を見る

    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年10月  [査読有り]

    DOI

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

    Trinh Khanh Duy

    LITHUANIAN MATHEMATICAL JOURNAL   51 ( 4 ) 486 - 506  2011年10月  [査読有り]

     概要を見る

    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
    被引用数
    (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年01月  [査読有り]

     概要を見る

    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
    被引用数
    (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年06月  [査読有り]

     概要を見る

    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
    被引用数
    (Scopus)

▼全件表示

共同研究・競争的資金等の研究課題

  • Universal approaches in random matrix theory

    日本学術振興会  科学研究費助成事業

    研究期間:

    2024年04月
    -
    2029年03月
     

    Trinh Khanh・Duy

  • Multi-aspects of beta ensembles and related random matrix models

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

    研究期間:

    2019年04月
    -
    2023年03月
     

    Trinh Khanh Duy

     概要を見る

    We study beta ensembles on the real line with focusing on the three classical beta ensembles (Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles). In a high temperature regime, we show a universality result at the bulk, that is, around any fixed reference energy, the local statistics converges in distribution to a homogeneous Poisson point process. For the three classical beta ensembles, we completely describe the global behavior, that is, two fundamental results on the convergence to a limit of the empirical distribution (law of large numbers) and Gaussian fluctuations around the limit (central limit theorem). We flexibly use tools from probability theory, spectral theory, theory of orthogonal polynomials and stochastic analysis. The limiting measure in a high temperature regime is related to associated Hermite polynomials (Gaussian case), associated Laguerre polynomials (Laguerre case) and associated Jacobi polynomials (Jacobi case).

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

    日本学術振興会  科学研究費助成事業

    研究期間:

    2016年04月
    -
    2019年03月
     

    Trinh Khanh・Duy, 中野 史彦

     概要を見る

    結合確率密度関数による直交/ユニタリ/斜交アンサンブルの自然な一般化ガウシアンベータアンサンブルは、独立成分を持つヤコビ行列と呼ばれる対称三重対角行列の固有値として実現されることが知られている.本研究では、ガウシアンベータアンサンブルについて、スペクトル測度・経験分布の収束やその極限近傍でのガウス的揺らぎなど、いくつかの新たなスペクトル特性を確立する.その研究手法は主にランダム行列モデルに基づいているが、広いクラスのランダムヤコビ行列にも適用可能である.

Misc

 

現在担当している科目

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学内研究所・附属機関兼任歴

  • 2022年
    -
    2024年

    理工学術院総合研究所   兼任研究員