Researcher Database

Akihito Hora
Faculty of Science Mathematics Mathematics

Researcher Profile and Settings


  • Faculty of Science Mathematics Mathematics

Job Title

  • Professor


  • (BLANK)(Kyoto University)

J-Global ID

Research Interests

  • functional analysis, probability theory   

Research Areas

  • Natural sciences / Basic analysis / Functional Analysis

Association Memberships

  • 日本数学会   

Research Activities

Published Papers

  • Akihito Hora
    PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 51 (4) 691 - 708 0034-5318 2015/12 [Refereed][Not invited]
    Concentration phenomena in statistical ensembles of Young diagrams have been investigated as static models first for the Plancherel ensemble by Vershik-Kerov and Logan-Shepp in 1970s and later in some other group-theoretical setting by Biane. On the other hand, a dynamical model of concentration for Young diagrams, which is not directly connected with group representations, was shown by Funaki-Sasada in the framework of hydrodynamic limit. The aim of this paper is to present a new dynamical model of concentration for Young diagrams featuring the group-theoretical sense. Starting from an initial state yielding concentration and a microscopic dynamics keeping the Plancherel measure invariant, we derive an evolution of the profiles of Young diagrams under a diffusive scaling limit. The resulting evolution along macroscopic time is described in terms of the notions of Voiculescu's free probability theory such as free compression and free convolution of Kerov transition measures.
  • Takeshi Hirai, Akihito Hora
    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 66 (4) 1191 - 1226 0025-5645 2014/10 [Refereed][Not invited]
    Let S be a finite group with a character, sgn, of order 2, and S' its central extension by a group Z = < z > of order 2. A representation pi of S' is called spin if pi (z sigma') = -pi(sigma') (sigma' is an element of S'), and the set of all equivalence classes of spin irreducible representations (= IRs) of S' is called the spin dual of S'. Take a finite number of such triplets (S-j', z(j), sgn(j)) (1 <= j <= m). We define twisted central product S' = S-1' (*) over cap S-2'...(*) over capS(m)' as a double covering of S = S-1 x...x S-m, S-j = S-j'/< z(j)>, and for spin IRs pi(j) of S-j', define twisted central product pi = pi(1)(*) over cap pi(2)(*) over cap...(*) over cap pi(m) as a spin IR of S'. We study their characters and prove that the set of spin IRs pi of this type gives a complete set of representatives of the spin dual of S', These results are applied to the case of representation groups S' for S = G(n), and 2L(n), and their (Frobenius-)Young type subgroups.
  • Akihito Hora, Takeshi Hirai
    Kyoto Journal of Mathematics 54 (4) 775 - 817 2156-2261 2014 [Refereed][Not invited]
    Adetailed study of the characters of script G∞(T), the wreath product of compact group T with the infinite symmetric group script G∞, is indispensable for harmonic analysis on this big group. In preceding works, we investigated limiting behavior of characters of the finite wreath product script Gn(T) as n → ∞ and its connection with characters of script G∞(T). This paper takes a dual approach to these problems. We study harmonic functions on double-struck Y(T), the branching graph of the inductive system of script Gn(T)'s, and give a classification of the minimal nonnegative harmonic functions on it. This immediately implies a classification of the characters of script G∞(T ), which is a logically independent proof of the one obtained in earlier works. We obtain explicit formulas forminimal nonnegative harmonic functions on double-struck Y(T) and Martin integral expressions for harmonic functions.
  • ヤング図形のエルゴード的な統計集団における集中現象
    洞 彰人
    数理解析研究所講究録 1825 75 - 90 2013 [Not refereed][Invited]
  • Representations of symmetric groups and asymptotic combinatorics
    Hora, A
    Sugaku Expositions 22 (1) 91 - 106 2009 [Refereed][Not invited]
  • Limits of characters of wreath products ${\mathfrak S}_n(T)$ of a compact group $T$ with the symmetric groups and characters of ${\mathfrak S}_\infty(T)$, I
    Hirai, T, Hirai, E, Hora, A
    Nagoya Math. J. 193 1 - 93 2009 [Refereed][Not invited]
  • Akihito Hora, Takeshi Hirai, Etsuko Hirai
    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 60 (4) 1187 - 1217 0025-5645 2008/10 [Refereed][Not invited]
    This paper is the second part of our study on limiting behavior of characters of wreath products (n)(T) of compact group T as n -> infinity and its connection with characters of infinity(T). Contrasted with the first part, which has a representation-theoretical flavor, the approach of this paper is based on probabilistic (or ergodic-theoretical) methods. We apply boundary theory for a fairly general branching graph of infinite valencies to wreath products of an arbitrary compact group T. We show that any character of infinity(T) is captured as a limit of normalized irreducible characters of (n)(T) as n -> infinity along a path on the branching graph of infinity(T). This yields reconstruction of an explicit chaxacter formula for infinity(T).
  • Akihito Hora, Nobuaki Obata
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 360 (2) 899 - 923 0002-9947 2008 [Refereed][Not invited]
    We propose the quantum probabilistic techniques to obtain the asymptotic spectral distribution of the adjacency matrix of a growing regular graph. We prove the quantum central limit theorem for the adjacency matrix of a growing regular graph in the vacuum and deformed vacuum states. The condition for the growth is described in terms of simple statistics arising from the strati. cation of the graph. The asymptotic spectral distribution of the adjacency matrix is obtained from the classical reduction.
  • Jucys-Murphy element and walks on modified Young graph
    Hora, A
    Banach Center Publ. 73 223 - 235 2006 [Refereed][Not invited]
  • Takeshi Hirai, Etsuko Hirai, Akihito Hora
    JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY 46 (1) 75 - 106 0023-608X 2006 [Refereed][Not invited]
    Characters of factor representations of finite type of the wreath products G = G(infinity)(T) of any compact groups T with the infinite symmetric group G(infinity). were explicitly given in [HH4]-[HH6], as the extremal continuous positive definite class functions f(A) on G determined by a parameter A. In this paper, we give a special kind of realization of a factor representation pi(A) associated to f(A). This realization is better than the Gelfand-Raikov realization pi f, f = f(A), in [GR] at least at the point where a matrix element (pi(A) (g) v(0), v(0)) of pi(A) for a cyclic-vector v(0) can be calculated explicitly, which is exactly equal to the character f(A) (and so pi(A) has a trace-element v(0)). So the positive-definiteness of class functions f(A) given in [HH4]-[HH6] is automatically guaranteed, a proof of which occupies the first half of [HH6] in the case of T infinite. The case where T is abelian contains the cases of infinite Weyl groups and the limits G(infinity) (Z(r)) = lim(n ->infinity) (r, 1, n) of complex reflexion groups.
  • 対称群の表現と漸近的組合せ論
    洞 彰人
    数学 57 (3) 242 - 254 2005 [Refereed][Invited]
  • Remark on Biane's character formula and concentration phenomenon in asymptotic representation theory
    Hora, A
    Infinite Dimensional Harmonic Analysis 3 141 - 159 2005 [Refereed][Not invited]
  • The limit shape of Young diagrams for Weyl groups of type B
    Hora, A
    Oberwolfach Reports 2 (2) 2005 [Refereed][Not invited]
  • An interacting Fock space with periodic Jacobi parameter obtained from regular graphs in large scale limit
    Hora, A, Obata, N
    Quantum Information 5 121 - 144 2005 [Refereed][Not invited]
  • Free Probability and Asymptotic Representation Theory of Symmetric Groups
    Hora, A
    数理解析研究所講究録 1418 10 - 40 2005 [Not refereed][Invited]
  • Hora, A
    Interdisciplinary Information Sciences 10 (1) 1 - 10 2004 [Refereed][Not invited]
  • A Hora
    INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS 6 (1) 139 - 143 0219-0257 2003/03 [Refereed][Not invited]
    Asymptotic behavior of spectral distribution of the adjacency operator on the Johnson graph with respect to the Gibbs state is discussed in infinite volume and zero temperature limit. The limit picture is drawn on the one-mode interacting Fock space associated with Meixner polynomials.
  • A noncommutative version of Kerov's Gaussian limit for the Plancherel measure of the symmetric group
    Hora, A
    Lecture Notes in Math. 1815 77 - 88 2003 [Refereed][Not invited]
  • Quantum decomposition and quantum central limit theorem
    Hora, A, Obata, N
    Quantum Probability and White Noise Analysis 17 284 - 305 2003 [Refereed][Not invited]
  • Y Hashimoto, A Hora, N Obata
    JOURNAL OF MATHEMATICAL PHYSICS 44 (1) 71 - 88 0022-2488 2003/01 [Refereed][Not invited]
    A new method is proposed for investigating spectral distribution of the combinatorial Laplacian (adjacency matrix) of a large regular graph on the basis of quantum decomposition and quantum central limit theorem. General results are proved for Cayley graphs of discrete groups and for distance-regular graphs. The Coxeter groups and the Johnson graphs are discussed in detail by way of illustration. In particular, the limit distributions obtained from the Johnson graphs are characterized by the Meixner polynomials which form a one-parameter deformation of the Laguerre polynomials (C) 2003 American Institute of Physics.
  • ヤング図形の極限形状とゆらぎにまつわる漸近的組合せ論
    洞 彰人
    数理解析研究所講究録 1310 85 - 104 2003 [Not refereed][Invited]
  • Noncommutative aspect of central limit theorem for the irreducible characters of the symmetric groups
    Hora, A
    Quantum Probability and White Noise Analysis 16 318 - 328 2002 [Refereed][Not invited]
  • 量子分解法による隣接作用素のスペクトル解析 I: 個数作用素が現れない場合
    尾畑伸明, 洞彰人
    数理解析研究所講究録 1291 11 - 44 2002 [Not refereed][Invited]
  • Gibbs state, quadratic embedding, and central limit theorem on large graphs
    Hora, A
    Quantum Information 3 67 - 74 2001 [Refereed][Not invited]
  • The symmetric groups and algebraic central limit theorems
    Hora, A
    数理解析研究所講究録 1227 145 - 153 2001 [Not refereed][Invited]
  • A Hora
    PROBABILITY THEORY AND RELATED FIELDS 118 (1) 115 - 130 0178-8051 2000/09 [Refereed][Not invited]
    On the adjacency algebra of a distance-regular graph we introduce an analogue of the Gibbs state depending on a parameter related to temperature of the graph. We discuss a scaling limit of the spectral distribution of the Laplacian on the graph with respect to the Gibbs state in the manner of central limit theorem in algebraic probability, where the volume of the graph goes to infinity while the temperature tends to 0. In the model we discuss here (the Laplacian on the Johnson graph), the resulting limit distributions farm a one parameter family beginning with an exponential distribution (which corresponds to the case of the vacuum state) and consisting of its deformations by a Bessel function.
  • Scaling limit of the spectral distributions of the Laplacians on large graphs
    Hora, A
    Infinite dimensional harmonic analysis 2 192 - 202 2000 [Refereed][Not invited]
  • An axiomatic approach to the cut-off phenomenon for random walks on large distance-regular graphs
    Hora, A
    Hiroshima Mathematical Journal 30 (2) 271 - 299 2000 [Refereed][Not invited]
  • A Hora
    COMMUNICATIONS IN MATHEMATICAL PHYSICS 195 (2) 405 - 416 0010-3616 1998/07 [Refereed][Not invited]
    An adjacency operator on a group is a formal sum of(left) regular representations over a conjugacy class. For such adjacency operators on the infinite symmetric group which are parametrized by the Young diagrams, we discuss the correlation of their powers with respect to the vacuum vector state. We compute exactly the correlation function under suitable normalization and through the infinite volume limit. This approach is viewed as a central limit theorem in quantum probability, where the operators are interpreted as random variables via spectral decomposition. In [K], Kerov showed the corresponding result for one-row Young diagrams. Our formula provides an extension of Kerov's theorem to the case of arbitrary Young diagrams.
  • A Hora
    INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS 1 (2) 221 - 246 0219-0257 1998/04 [Refereed][Not invited]
    Regarding the adjacency matrix of a graph as a random variable in the framework of algebraic or noncommutative probability, we discuss a central limit theorem in which the size of a graph grows in several patterns. Various limit distributions are observed for some Cayley graphs and some distance-regular graphs. To obtain the central limit theorem of this type, we make combinatorial analysis of mixed moments of noncommutative random variables on one hand, and asymptotic analysis of spectral structure of the graph on the other hand.
  • Central limit theorem related to the correlation of conjugacy classes in the infinite symmetric group
    Hora, A
    数理解析研究所講究録 1035 104 - 113 1998 [Not refereed][Invited]
  • A Hora
    PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 33 (4) 695 - 710 0034-5318 1997/12 [Refereed][Not invited]
    The cut-off phenomenon is a remarkable critical phenomenon observed in the process of convergence to equilibrium in a wide variety of Markov chains. Diaconis-Graham-Morrison [3] established the first precise evaluation around the critical time for Ehrenfests' urn model concerning 2 urns and d balls, i.e. nearest neighbor random walks on hypercube Z(2)(d). They showed that deviation from the equilibrium state is described well by using the error function. In this article, we work out the evaluation around the critical time for simple random walks on Hamming graphs H(d,n), which coincide with an extended Ehrenfests' urn model concerning n urns and d balls. In our case, not only d but also n can grow in several manners. If n/d tends to 0, the similar result to [3] remains valid and microscopic deviation from the equilibrium state is described by the error function. If n/d tends to a nonzero constant, however, it is shown that the error function has to be replaced by an expression involving Poisson distributions.
  • A critical phenomenon appearing in the process of particle diffusion in classical statistical mechanics
    Hora, A
    Journal of Faculty of Environmental Science and Technology, Okayama University 2 1 - 8 1997 [Not refereed][Not invited]
  • ランダムウォークのカットオフ現象
    洞 彰人
    数理解析研究所講究録 1017 70 - 91 1997 [Not refereed][Invited]
  • Towards critical phenomena for random walks on various algebraic structures
    Hora, A
    Infinite dimensional harmonic analysis 1 113 - 127 1996 [Refereed][Not invited]
  • Random walks and isotropic Markov chains on homogeneous spaces
    Hora, A
    Journal of Faculty of Environmental Science and Technology, Okayama University 1 21 - 26 1996 [Not refereed][Not invited]
  • 量子確率論とグラフのスペクトル解析
    洞 彰人
    数理解析研究所講究録 957 109 - 121 1996 [Not refereed][Invited]
  • The cut-off phenomenon in random walks on association schemes
    Hora, A
    数理解析研究所講究録 962 32 - 41 1996 [Not refereed][Invited]
  • 量子ランダムウォークに関する話題
    洞 彰人
    数理解析研究所講究録 923 124 - 138 1995 [Not refereed][Invited]
  • リー環の表現のテンソル積の分解から生じるランダムウォークに ついて
    洞 彰人
    数理解析研究所講究録 887 169 - 179 1994 [Not refereed][Invited]
  • A HORA
    PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 29 (1) 153 - 159 0034-5318 1993/03 [Refereed][Not invited]
    The shuffling problem is discussed as the asymptotic behavior of random walks on finite groups. We give a new characterization for asymptotic equidistribution of such random walks in terms of representations of the group. As applications, we characterize perfect groups and consider random walks on classical Weyl groups.
  • Akihito Hora
    Journal of Theoretical Probability 5 (1) 71 - 100 0894-9840 1992/01 [Refereed][Not invited]
    Investigated is quasi-invariance of power probabilities on the infinite product of SU(2). We consider the subgroup consisting of those actions which keep a measure quasi-invariant (i.e., mutually absolutely continuous) and call it the quasi-invariant subgroup of the measure. We establish several estimations for the quasi-invariant subgroups in terms of lfp-type subgroups of SU(2)∞. Our methods are based on Hellinger integrals, Fourier analysis, and spherical functions on SU(2). © 1992 Plenum Publishing Corporation.
  • A HORA
    MATHEMATISCHE ZEITSCHRIFT 206 (2) 169 - 192 0025-5874 1991 [Refereed][Not invited]
  • A HORA
    PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 24 (5) 739 - 757 0034-5318 1988/12 [Refereed][Not invited]
    PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 23 (2) 275 - 296 0034-5318 1987/03 [Refereed][Not invited]
  • ヤング図形集団における極限形状とガウスゆらぎの動的モデル
    洞 彰人
    数理解析研究所講究録 [Not refereed][Invited]

Books etc

  • 対称群の表現とヤング図形集団の解析学--漸近的表現論への序説--
    洞 彰人 (Single work)
    数学書房 2017
  • The Limit Shape Problem for Ensembles of Young Diagrams
    Hora, A (Single work)
    Springer 2016
  • Projective representations and spin characters of complex reflection groups $G(m,p,n)$ and $G(m,p,\infty)$
    Hirai, T, Hora, A, Hirai, E (Joint work)
    Mathematical Society of Japan 2013
  • この定理が美しい, ランダムネスに潜む普遍性---中心極限定理
    洞 彰人 (Contributor176--185)
    数学書房 2009
  • Infinite Dimensional Harmonic Analysis
    Hilgert, J, Hora, A, Kawazoe, A, Nishiyama, K, Voit, M (Joint editor)
    World Scientific 2008
  • Quantum Probability and Spectral Analysis of Graphs
    Hora, A, Obata, N (Joint work)
    Springer 2007
  • Non-Commutativity, Infinite Dimensionality and Probability at the Crossroads
    Obata, N, Matsui, T, Hora, A (Joint editor)
    World Scientific 2002

Conference Activities & Talks

  • Dynamical scaling limit of the restriction-induction chain on Young diagrams in terms of free probability  [Invited]
    HORA Akihito
    Random Matrices and Their Applications  2018/05
  • Markov chains, graph spectra, and some static/dynamic scaling limits  [Invited]
    HORA Akihito
    第3回代数的組合せ論「仙台勉強会」  2018/03
  • On evolution of macroscopic profiles (and their global fluctuations) for growing random Young diagrams  [Invited]
    HORA Akihito
    第19回北東数学解析研究会  2018/02
  • 群論的なヤング図形集団における巨視的プロファイルとゆらぎの動的モデル  [Invited]
    洞 彰人
    筑波大学解析セミナー  2017/11
  • Dynamic model for limit profiles and their Gaussian fluctuations in Young diagram ensembles  [Invited]
    HORA Akihito
    Mathematical Aspects of Quantum Fields and Related Topics  2017/06
  • On a dynamic model for limit profiles and their Gaussian fluctuations in group-theoretical ensembles of Young diagrams  [Invited]
    HORA Akihito
    Colloquium RIMS Kyoto Univ.  2017/05
  • Application of free probability to dynamical limit shapes of random Young diagrams  [Not invited]
    HORA Akihito
    One-day workshop on Interface between Commutative and Non-Commutative Stochastic Analysis  2017/03
  • ヤング図形集団における大数の法則(静的および動的モデル)  [Invited]
    洞 彰人
    岡山-広島解析・確率論セミナー  2017/02
  • 制限誘導連鎖の流体力学極限と自由確率論  [Not invited]
    洞 彰人
    日本数学会年会  2015/03
  • プランシェレル集団における流体力学極限と自由確率論  [Invited]
    洞 彰人
    札幌数理物理研究集会  2014/09
  • 対称群の表現の漸近理論への誘い  [Invited]
    洞 彰人
    非可換解析集中セミナー, 愛知教育大学  2013/09
  • Growth process of multi-diagrams, its Martin boundary, and characters of an inductive limit group  [Invited]
    HORA Akihito
    Markov Chains on Graphs and Related Topics, RIMS International Project Research 2012: Discrete Geometric Analysis  2013/02
  • 帰納極限群の分岐グラフ上の調和関数  [Invited]
    洞 彰人
    月曜解析セミナー, 北海道大学  2012/11
  • コンパクト群の帰納系を舞台にした確率論と調和解析の話題  [Invited]
    洞 彰人
    北大数学談話会  2012/10
  • ヤング図形のエルゴード的な統計集団における集中現象  [Invited]
    洞 彰人
    表現論と非可換調和解析の展望, 京都大学数理解析研究所  2012/06
  • コンパクト群の環積の指標と分岐グラフの境界  [Not invited]
    洞 彰人
    日本数学会年会  2012/03
  • ヤング図形の群論的統計集団における最尤形状について  [Invited]
    洞 彰人
    岡山解析・確率セミナー  2011/10
  • Harmonic functions on branching networks for some big groups  [Invited]
    HORA Akihito
    Sapporo Workshop on Non-commutative Analysis and Applications to Complex Phenomena  2011/09
  • ヤンググラフ上の調和関数と無限対称群の表現  [Invited]
    洞 彰人
    離散幾何解析セミナー  2011/07
  • Characters and harmonic functions related to infinite wreath product groups  [Invited]
    HORA Akihito
    RIMS Project Research, The International Conference on Functions in Number Theory and Their Probabilistic Aspects  2010/12


  • Joint Research on Infinite-Dimensional Harmonic Analysis
    2000 -2001
  • Joint Research on Algebraic Probability


  • 群の表現論と極限定理 --- オルシャンスキーとオクニコフ
    洞 彰人  数理科学  20  -25  2008/12  [Refereed][Invited]
  • ヤング図形の漸近挙動をめぐる調和解析と確率論の話題
    洞 彰人  日本数学会年会企画特別講演アブストラクト  2007  [Not refereed][Invited]
  • オクニコフ(フィールズ賞業績紹介)
    洞 彰人  数学セミナー  2006/01  [Refereed][Invited]

Research Grants & Projects

  • 漸近的表現論の深化と展開
    Author : 洞 彰人
  • Harmonic Analysis, Random Walks, Noncommutative Probability
  • 巨大な群上の調和解析に向けた確率論と表現論の融合的研究
    Author : 洞 彰人
  • 群のユニタリ表現の分解と分岐グラフ上の調和関数の研究
    Author : 洞 彰人

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