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Analysis of Critical Phenomena in Random Systems

Research Project

Project/Area Number 09440079
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKobe University

Principal Investigator

HIGUCHI Yasunari  Kobe University, Faculty of Science, Professor, 理学部, 教授 (60112075)

Co-Investigator(Kenkyū-buntansha) WATANABE Kiyoshi  Kobe University, Faculty of Science, Associate Professor, 理学部, 助教授 (60091245)
TAKANO Kyoichi  Kobe University, Faculty of Science, Professor, 理学部, 教授 (10011678)
FUKUYAMA Katusi  Kobe University, Faculty of Science, Associate Professor, 理学部, 助教授 (60218956)
KONNO Norio  Yokohama National University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (80205575)
YOSHIDA Nobuo  Kyoto University, Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (40240303)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥11,100,000 (Direct Cost: ¥11,100,000)
Fiscal Year 1999: ¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 1998: ¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 1997: ¥4,000,000 (Direct Cost: ¥4,000,000)
KeywordsGlauber dynamics / spectral gap / critical point / Ising model / percolation / shrink of contours / reinforced random walk / コントゥアーの収縮 / First passage percolation / fluctuation / コントゥアー / 境界条件 / Ising model / Glauber dynamics / spectral gap / reinforced random walk / 再帰性 / 向きつきパーコレーション / 相関不等式
Research Abstract

We investigated the spectral gap of Glauber dynamics in a finite square in the low temperature case in two dimensions. This is known to have a uniform lower bound independent of the boundary condition when the temperature is higher than the critical point, and goes to zero as the size of the square goes to infinity in the low temperature case.
We obtained a sharpe lower bound of the spectral gap for the + boundary condition up to the critical point : gap(L,+)【greater than or equal】exp{-CィイD8LlogLィエD8}, where L is the size of the box. This bound is better than the existing one. It is even better than the estimate which is known only for the low enough temperature.
Further, we showed that the uniform spectral gap is equivalent to the strong mixing condition even for unbounded spin systems.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (84 results)

All Other

All Publications (84 results)

  • [Publications] Y. Higuchi 他: "On the speed of convergence for two dimensional first-passage percolation"The Annals of Probability. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.O. Georgii 他: "Percolation and number of phases in the 2D Ising model"Journal of Mathematical Physics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Higuchi: "Level set representation for Gibbs states and magnetic field processes"Trends in Probability and related Analysis, N. Kono & N.R. Shieh, World Scientific. 167-175 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Higuchi: "Slow relaxation of 2-D stochastic Ising models with random and non-random boundary conditions"New trends in Stochastic Analysis, Proc. Taniguchi Workshop, Charingworth Manor, World Scientific. 153-167 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Higuchi: "Ising percolation"Sugaku Expositions. 10. 143-158 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Fukuyama: "The central limit theorem for Σf(θ^nx)g(θ^<n2>x)"Ergodic Theory and Dynamical Systems. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Fukuyama: "On the gap series of Mark Kac"Colloquium Mathematicum. 81. 157-160 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Fukuyama 他: "The central limit theorem for lacunary series"Proc. Amer. Math. Soc.. 127. 599-608 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Fukuyama 他: "A remark on the g-CLT result by Molgenthaler"Trends in Probability and related Analysis, eds. N. Kono & N.R. Shieh, World Scientific. 151-158 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Fukuyama: "Riesz-Raikov sums and irrational rotations"Trends in Probability and related Analysis, eds. N. Kono & N.R. Shieh, World Scientific. 151-158 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Takano: "Defining manifolds of Painleve equations"Toward the exact WKB analysis of differential equations linear or nonlinear (Eds. T. Kawai and Y. Takei). (印刷中). (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Matano 他: "On some Hamiltonian structure of Painleve systems, II"Journal of Mathematical Society of Japan. 51. 843-866 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Shioda 他: "On some Hamiltonian structure of Painleve systems, I"Funkcialaj Ekvacioj. 40. 843-866 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Watanabe: "Weierstrass points of the Fermat curve"Proceedings of the Congress ISAAC '97, University of Delaware, U.S.A., 3-7 June, 1997, Kluwer Academic. 1-13 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Watanabe: "A note on weights of Weierstrass points"Mathematics Journal of Toyama University. 2. 117-120 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Watanabe: "Cousin problems and holomorphic cocycles"Mathematics Japonica. 48. 217-221 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Sekiguchi: "Combinatorial formula of the dimension of global solutions to a generalized hypergeometric system 【〜!M】_<3,2>(ν)"Tokyo Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Katori 他: "Survival probabilities for discrete time models in one dimension"Journal of Statistical Physics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Kanno 他: "Lower bounds for critical values of a cancellative model"Journal of Physics A : Mathematical and General. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Takahashi 他: "Positive correlation inequalities for nonattractive stochastic cellular automata"Proceedings of International Workshop on Soft Computing in Industry '99. 474-478 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Inui 他: "Local directed percolation probability in two dimensions"Journal of the Physical Society of Japan. 67. 99-102 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Konno: "Upper bounds on survival probabilities for a nonattractive model"Journal of the Physical Society of Japan. 66. 3751-3755 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] V. Belitsky 他: "A strong correlation inequality for contact processes and oriented percolation"Stochastic Processes and their Applications. 67. 213-225 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Yoshida: "Application of log-Sobolev inequality to the stochastic dynamics of unbounded spin systems on the lattice"J. Functional Analysis. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Yoshida: "The log-Sobolev inequality for weakly coupled lattice fields"Probability Theory and Related Fields. 115. 1-40 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Yoshida: "Finite volume Glauber dynamics in a small magnetic field"Journal of Statistical Physics. 90. 1015-1035 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. Schonmann 他: "Exponential relaxation of Glauber dynamics with some specia boundary conditions"Communications in Mathematical Physics. 189. 299-310 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Yoshida: "Exponential relaxation of finite volume Glauber dynamics near the border of the one phase region"Trends in Probability and Related Analysis, ed. by N. Kono and N.-R. Shieh, World Scientific. 339-350 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Yoshida: "Relaxed criteria of the Dobrushin-Shlosmn mixing condition"Journal of Statistical Physics. 87. 293-309 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] J. Murai: "Percolation on high dimensional Menger sponges"Kobe Journal of Mathematics. 14. 49-61 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Higuchi and others: "On the speed of convergence for two dimensional firstpassage percolation"The Annals of Probability. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.O.Georgii and others: "Percolation and number of phases in the 2D Ising model"Journal of Mathematical Physics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Higuchi: "Level set representation for Gibbs states and magnetic field processes"Trends in Probability and related Analysis, N. Kono & N. R. Shieh, World Scientific. 167-175 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Higuchi and others: "Slow relaxation of 2-D stochastic Ising models with random and non-random boundary conditions"New trends in Stochastic Analysis, Proc. Taniguchi Workshop, Charingworth Manor, World Scientific. 153-167 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Higuchi: "Ising percolation"Sugaku Expositions. 10. 143-158 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Fukuyama: "The central limit theorem for Σf(θィイD1nィエD1x)g(θィイD1nィエD1ィイD12ィエD1x)"Ergodic Theory and Dynamical Systems. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Fukuyama and others: "The central limit theorem for lacunary series"Proc. Amer. Math. Soc.. 127. 599-608 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Fukuyama and others: "A remark to the g-CLT result by Morgenthaler"Trends in Probability and related Analysis, eds. N. Kono & N. R. Shieh, World Scientific. (in press). (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Fukuyama: "Riesz-Raikov sums and irrational rotations"Trends in Probability and related Analysis, eds. N. Kono & N. R. Shieh, World Scientific. 151-158 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Takano: "Difining manifolds for Painleve equations"Toward the exact WKB analysis of differential equations linear or nonlinear(Eds. T. Kawai and Y. Takei). (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Matano and others: "On some Hamiltonian structures of Painleve systems, II"Journal of Mathematical Society of Japan. 51. 843-866 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Shioda and others: "On some Hamiltonian structures of Painleve systems, I"funkcialaj Ekvacioj. 40. 271-291 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Watanabe: "Weierstrass points of the Fermat curve"Proceedings of the Congress ISAAC'97, University of Delaware, U.S.A, 3-7 June, 1997, Kluwer Academic. 1-13 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Watanabe: "A note on weights of Weierstrass points"Mathematics Journal of Toyama University. 2. 117-120 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Watanabe: "Cousin problems and holomorphic cocycles"Mathematica Japonica. 217-221 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Sekiguchi: "Combinatorial formula of the dimension of global solutions to a generalized hypergeometric system MィイD4〜ィエD4ィイD23,2ィエD2(v)"Tokyo Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Katori and others: "Survival probabilities for discrete time models in one dimension"Journal of Statistical Physics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N.Konno and others: "Lower bounds for critical values of a cancellative model"Journal of Physics A : Mathematical and General. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Takahashi and others: "Positive correlation inequalities for nonattractive stochastic cellular automata"Proceedings of International Workshop on Soft Computing in Industry'99. 474-478 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N.Inui and others: "Local directed percolation probability in two dimensions"Journal of the Physical Society of Japan. 61. 99-102 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N.Konno: "Upper bounds on survival probabilities for a nonattractive model"Journal of the Physical Society of Japan. 66. 3751-3755 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] V.Belitsky and others: "A strong correlation inequality for contact processes and oriented percolation"Stochastic Processes and their Applications. 67. 213-225 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N.Yoshida: "Application of log-Sobolev inequality to the stochastic dynamics of unbounded spin system on the lattice"J. Functional Analysis. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N.Yoshida: "The log-Sobolev inequality for weakly coupled lattice fields"Probability Theory and Related Fields. 115. 1-40 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N.Yoshida: "Finite volume Glauber dynamics in a small magnetic field"Journal of Statistical Physics. 90. 1015-1035 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R.Schonmann and others: "Exponential relaxation of Glauber dynamics with some specia boundary conditions"Communications in Mathematical Physics. 189. 299-310 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N.Yoshida: "Exponential relaxation of finite volume Glauber dynamics near the border of the one phase region"Trends in Probability and Related Analysis, ed. by N. Kono and N. -R. Shieh, World Scientific. 339-350 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] J.Murai: "Percolation on high dimensional Menger sponges"Kobe Journal of Mathematics. 14. 49-61 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Higuchi 他: "On the speed of convergence for two dimensional first-passage percolation"The Annals of Probability. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] H.O.Georgii 他: "Percolation and number of phases in the 2D Ising model"Journal of Mathematical Physics. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Fukuyama: "On the gap series of Mark Kac"Colloquium Mathematicum. 81. 157-160 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Fukuyama: "The central limit theorem for lacunary series"Proc.Amer.Math.Soc.. 127. 599-608 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Fukuyama 他: "A remark on the g-CLT result by Molgenthaler"Trends in Probability and related Analysis,eds.N.Kono & N.R.Shieh,World Scientific,. 203-206 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Fukuyama: "The central limit theorem for Σf(θ^nx)g(θ^<n2>x)"Ergodic Theory and Dynamical Systems. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Matano 他: "On some Hamiltonian structures of Painleve systems,II"Journal of Mathematical Socienty of Japan. 51. 843-866 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Takano: "Difining manifolds for Painleve equations"Toward the exact WKB analysis of differential equations linear or nonlinear(Eds.T.Kawai and Y.kakei). (印刷中). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Watanabe: "Weierstrass points of the Fermat curve"Proceedings of the Congress ISSA'97,University of Delaware,U.S.A,3-7 June,1997,Kluwer Academic. 1-13 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Sekiguchi: "Combinatorial formula of the dimension of global soulutions to a generalized hypergeometric system M_<3,2>(v)"Tokyo Journal of Mathematics. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Takahashi 他: "Positive correlation inequalities for nonattractive stochastic cellular automata"Proceedings of International Workshop on Soft Computing inIndustry '99. 474-478 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Katori 他: "Suvival probabilities for discrete time models in one dimension"Journal of Statistical Physics. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] N.Konno 他: "Lower bounds for critical values of a cancellative model"Journal of Physics A: Mathematical and General. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] N.Yoshida: "The log-Sobolev inequality for weakly coupled lattice fields"Probability Theory and Related Fields. 115. 1-40 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] N.Yoshida: "Application of log-Sobolev inequality to the stochastic dynamics of unbounded spin systems on the lattice"J.Functional Analysis. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Higuchi: "Level set reresentation for gibbs states and magnetic field processes" Trnds in Probability and related Analysis, eds. N.Kono & N.R.Shieh,World Scientific,. 167-175 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Fukuyama: "The central limit theorem for lacunary series" Proc.Amer.Math.Soc.127. 599-608 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Yoshida: "Finite volume Glauber dynamics in a small magnetic fild" J.Stat.Phys.90. 1015-1035 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 今野紀雄: "無限粒子系の数理-コンタクトプロセスの相転移現象" トライボロジスト. 43. 696-701 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 関口英子: "古典型有界対称領域における Penrose 変換の高次元化と特異ユニタリ表現" リー群と幾何学シンポジウム講演要旨. 1-8 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Higuchi and N.Yoshida: "Slow relaxation of stochastic Ising models with random and non-random boundary conditions," In : New trends in stochastic analysis, K.D.Elworthy, S.Kusuoka, I.Shigekawa (eds), World Scientific Publishing. 153-167 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Higuchi: "Level set representation for Gibbs states and magnetic field processes" In : Trends in probability and related analysis, N.Kono and N-R. Shieh (eds). 167-175 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Fukuyama: "Riesz-Raikov sums and irrational rotations," in: Trends in Probability and related Analysis, N.Kono and N-R. Shieh (eds). 151-158 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] R.Schonmann and N.Yoshida: "Exponential relaxation of Glauber dynamics with some special boundary conditions," Commun.Math.Phys.189. 299-310 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] V.Belitsky, N.Konno et.al: "A strong correlation inequality for contact processes and oriented percolation" Stochastic Processes and their Applications. 167. 213-225 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Shioda and K.Takano: "On Some Hamiltonian Structures on Pianleve Systems, I" Funkcial.Ekvac.40. 271-310 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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