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The relationship between the behaviour of random walks and the spectral structure via some combinatorial geometries of graphs

Research Project

Project/Area Number 25400208
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionShowa University

Principal Investigator

HIGUCHI Yusuke  昭和大学, 教養部, 講師 (20286842)

Research Collaborator SEGAWA Etsuo  
NOMURA Yuji  
OGURISU Osamu  
Sato Iwao  
KONNO Norio  
ANDO Kazunori  
MORIOKA Hisashi  
SUZUKI Akito  
TANAKA Mikihiro  
Portugal Rento  
Project Period (FY) 2013-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥5,200,000 (Direct Cost: ¥4,000,000、Indirect Cost: ¥1,200,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywordsグラフ理論 / スペクトル幾何 / ラプラシアン / 酔歩 / 量子ウォーク / 状態密度函数 / 被覆構造 / 脳内辞書ネットワーク / 離散スペクトル幾何 / 定常測度 / 共鳴状態
Outline of Final Research Achievements

Let us show three topics in our research supported by this grant. One is about quantum walks: we showed the spectral relationship between quantum walk and its underlying random walk. This kind of relation is now called “Spectral mapping theorem in quantum walk” and widely applied in this area. Next one is about some resonance of discrete Laplacians on a graph. We can characterize the intensities of potentials to keep the existence of edge-eigenvalues, which implies some eigenvalues whose value coincides with the lowest of continuous spectrum. Moreover we succeeded in construction of algebraic variety composed by such intensites. Finally, we tried to apply our basic results to one of other fields than mathematics: mental lexicon, which is one of topics in psycholinguistics.

Academic Significance and Societal Importance of the Research Achievements

酔歩がより速く伝播するグラフの特徴付けは,未だ未解決な部分が多い双曲的なグラフの解明,さらには現実社会におけるネットワークの構築に貢献できるものである.一方,酔歩よりも速い拡散性を持つ量子ウォークへの研究成果は,今後より現実的に扱われるだろう量子コンピューティングなどの“量子化”された媒体での実装に貢献できると考えている.つまり当該研究は現状と未来を見据えた意義をもたらすと期待しても良いだろう.

Report

(7 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • 2013 Research-status Report
  • Research Products

    (23 results)

All 2018 2017 2016 2015 2014 2013 Other

All Int'l Joint Research (1 results) Journal Article (10 results) (of which Peer Reviewed: 10 results,  Open Access: 3 results,  Acknowledgement Compliant: 4 results) Presentation (12 results) (of which Invited: 10 results)

  • [Int'l Joint Research] LNCC(ブラジル)

    • Related Report
      2018 Annual Research Report
  • [Journal Article] Quantum walks induced by Dirichlet random walks on infinite trees2018

    • Author(s)
      Yusuke HIGUCHI and Etsuo SEGAWA
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 51 Issue: 7 Pages: 075303-075303

    • DOI

      10.1088/1751-8121/aa8fba

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Periodicity of the Discrete-time Quantum Walk on a Finite Graph2017

    • Author(s)
      Yusuke Higuchi, Norio Konno, Iwao Sato and Etsuo Segawa
    • Journal Title

      Interdisciplinary Information Sciences

      Volume: 23 Issue: 1 Pages: 75-86

    • DOI

      10.4036/iis.2017.A.10

    • NAID

      130005519657

    • ISSN
      1340-9050, 1347-6157
    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] THE SPREADING BEHAVIOR OF QUANTUM WALKS INDUCED BY DRIFTED RANDOM WALKS ON SOME MAGNIFIER GRAPH2017

    • Author(s)
      Yusuke HIGUCHI and Etsuo SEGAWA
    • Journal Title

      Quantum Information and Computation

      Volume: 17

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the discrete eigenvalues of a discrete Schroedinger operator with a finitely supported potential2016

    • Author(s)
      Yusuke Hayashi, Yusuke Higuchi, Yuji Nomura and Osamu Ogurisu
    • Journal Title

      Letters in Mathematical Physics

      Volume: 106 Issue: 11 Pages: 1465-1478

    • DOI

      10.1007/s11005-016-0876-0

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] A Balanced Signed Digraph2015

    • Author(s)
      Yusuke Higuchi and Iwao Sato
    • Journal Title

      Graphs and Combinatorics

      Volume: 31 Issue: 6 Pages: 2215-2230

    • DOI

      10.1007/s00373-014-1496-z

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] A remark on zeta functions of finite graphs via quantum walks2014

    • Author(s)
      Yusuke HIGUCHI, Norio KONNO, Iwao SATO and Etsuo SEGAWA
    • Journal Title

      Pacific Journal of Mathematics for Industry

      Volume: 6 Issue: 1

    • DOI

      10.1186/s40736-014-0009-6

    • NAID

      40020377848

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Spectral and asymptotic properties of Grover walks on crystal lattice2014

    • Author(s)
      Yusuke Higuchi, Norio Konno, Iwao Sato and Etsuo Segawa
    • Journal Title

      Journal of Functional Analysis

      Volume: 267 Issue: 11 Pages: 4197-4235

    • DOI

      10.1016/j.jfa.2014.09.003

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Quantum graph walks I: mapping to quantum walks2013

    • Author(s)
      Yusuke HIGUCHI, Norio KONNO, Iwao SATO and Etsuo SEGAWA
    • Journal Title

      Yokohama Math. J.

      Volume: 59

    • NAID

      120006360264

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Quantum graph walks II: quantum walks on covering graphs2013

    • Author(s)
      Yusuke HIGUCHI, Norio KONNO, Iwao SATO and Etsuo SEGAWA
    • Journal Title

      Yokohama Math. J.

      Volume: 59

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] A note on the discrete-time evolutions of quantum walk on a graph.2013

    • Author(s)
      Yusuke HIGUCHI, Norio KONNO, Iwao SATO and Etsuo SEGAWA
    • Journal Title

      Journal of Math-for-Industry

      Volume: 5(2013B-3)

    • NAID

      120005372053

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Presentation] グラフの幾何がもたらす作用素のスペクトル構造2018

    • Author(s)
      樋口 雄介
    • Organizer
      RIMS共同研究「量子ウォークと場の量子論における超対称性の数理」
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Combinatorial structure in the dictionary and the mental lexicon2018

    • Author(s)
      樋口 雄介
    • Organizer
      2019 Workshop on Text Mining and Discrete Spectral Analysis
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Mental lexicon と Erdos-Renyi random graph model2018

    • Author(s)
      樋口 雄介
    • Organizer
      2018 Workshop on Text Mining and Discrete Spectral Analysis
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 文書分類に向けた点過程とスペクトル幾何2017

    • Author(s)
      樋口 雄介
    • Organizer
      2017 Workshop on Text Mining and Discrete Spectral Analysis
    • Place of Presentation
      愛媛大学工学部
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 酔歩と量子ウォークのスペクトル幾何2017

    • Author(s)
      樋口 雄介
    • Organizer
      2017 鹿児島スペクトル幾何学研究会
    • Place of Presentation
      鹿児島大学理学部
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 1次元 Szegedy walk の定常測度2015

    • Author(s)
      樋口 雄介
    • Organizer
      横浜国立大学理工学部 応用数学研究室セミナー
    • Place of Presentation
      横浜国立大学理学研究棟
    • Year and Date
      2015-11-24
    • Related Report
      2015 Research-status Report
  • [Presentation] An introduction to quantum walks via random walks on graphs2015

    • Author(s)
      Yusuke HIGUCHI
    • Organizer
      Workshop: Quantum walks, random walks and scattering theory on periodic systems (0)
    • Place of Presentation
      Graduate school of mathematics, Nagoya University
    • Year and Date
      2015-01-21 – 2015-01-22
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Zeta functions of finite graphs via quantum walks2014

    • Author(s)
      Yusuke HIGUCHI
    • Organizer
      第5回「ハミルトン系とその周辺」研究集会
    • Place of Presentation
      金沢大学サテライトプラザ
    • Year and Date
      2014-05-31
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] 離散スペクトル解析に寄添ふグラフ理論の翳(其の壱)(其の弐)2014

    • Author(s)
      樋口 雄介
    • Organizer
      2014 スペクトル・散乱 鹿児島シンポジウム
    • Place of Presentation
      鹿児島大学理学部
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] グラフ上の酔歩と量子ウォーク2014

    • Author(s)
      樋口 雄介
    • Organizer
      近畿大学数学教室講演会
    • Place of Presentation
      近畿大学理工学部理学科
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 量子ウォークが生みだすグラフのゼータ函数の変種2014

    • Author(s)
      樋口 雄介
    • Organizer
      田澤新成先生ご退職記念研究集会
    • Place of Presentation
      近畿大学 東大阪キャンパス
    • Related Report
      2013 Research-status Report
  • [Presentation] グラフ上の酔歩と量子ウォーク2013

    • Author(s)
      樋口 雄介
    • Organizer
      数理工学談話会
    • Place of Presentation
      熊本大学工学部
    • Related Report
      2013 Research-status Report
    • Invited

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Published: 2014-07-25   Modified: 2020-03-30  

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