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Rigorous analysis for high-dimensional critical behavior and crossover phenomena in mathematical models

Research Project

Project/Area Number 18K03406
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12040:Applied mathematics and statistics-related
Research InstitutionHokkaido University

Principal Investigator

Sakai Akira  北海道大学, 理学研究院, 教授 (50506996)

Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2020: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords自己回避歩行 / パーコレーション / イジング模型 / 相転移 / 臨界現象 / 上部臨界次元 / クロスオーバー現象 / レース展開 / 臨界2点関数
Outline of Final Research Achievements

(1) We prove that bond percolation on the d-dimensional body-centered cubic lattice exhibits mean-field behavior as soon as d≧9. Although it is still away from the expected bound d≧7,it is superior to the best-ever bound d≧11 for bond percolation on the d-dimensional hypercubic lattice, proven by Fitzner and van der Hofstad (2017).
(2) We prove that sufficiently spread-out long-range models of self-avoiding walk, percolation and the Ising model, which are defined by power-law decaying 2-body interactions (characterized by an exponent a), exhibit mean-field behavior as soon as d is bigger than the model-dependent upper-critical dimension dc when a≠2, and as soon as d≧dc when a=2. This solves a conjecture in physics (2014).

Academic Significance and Societal Importance of the Research Achievements

(1)或る次元dcより上では,臨界現象が平均場的なものに退化する.このdcを上部臨界次元dcと呼ぶ.最近接パーコレーションではdc=6と予想されているが,超立方格子上ではdc≦10であることが示されていた.本研究は,標準的な最近接格子の一つである体心立方格子上で一桁の次元dc≦8まで降りることに成功した初めての結果である.
(2)境界冪長距離モデルの場合,臨界2点関数の漸近挙動には(高次元でも)対数補正がつくものと予想されていたが,それを証明する解析手法がなかった.本研究では,対数補正つき冪関数どうしの畳み込み不等式を開発し,dc直上まで平均場臨界現象に退化することを証明した初めての結果である.

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (44 results)

All 2023 2022 2021 2020 2019 2018 Other

All Int'l Joint Research (13 results) Journal Article (7 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 7 results,  Open Access: 3 results) Presentation (16 results) (of which Int'l Joint Research: 11 results,  Invited: 13 results) Book (1 results) Remarks (4 results) Funded Workshop (3 results)

  • [Int'l Joint Research] University of Durham(英国)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] National Center for Theoretical Sciences(その他の国・地域)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] National Chengchi University/National Center for Theoretical Sciences/National Yang Ming Chiao Tung University(その他の国・地域)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] Durham University(英国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] University of Durham(英国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] National Chiao Tung University(台湾)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] Ludwig-Maximilians-Universitat Munchen(ドイツ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] University of British Columbia(カナダ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] National Chengchi University(その他の国・地域(台湾))

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] University of Bristol(英国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] National Chengchi University (Taiwan)(その他の国・地域)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] University of Bristol(英国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Ludwig-Maximilians Universitat Munchen(ドイツ)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Correct bounds on the Ising lace-expansion coefficients2022

    • Author(s)
      Akira Sakai
    • Journal Title

      Communications in Mathematical Physics

      Volume: - Issue: 3 Pages: 783-823

    • DOI

      10.1007/s00220-022-04354-5

    • Related Report
      2022 Annual Research Report 2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] パーコレーションの数理 20202022

    • Author(s)
      坂井 哲
    • Journal Title

      数学

      Volume: 74 Pages: 253-279

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] パーコレーションの数理20202022

    • Author(s)
      坂井 哲
    • Journal Title

      数学

      Volume: -

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Crossover phenomena in the critical behavior for long-range models with power-law couplings2020

    • Author(s)
      Akira Sakai
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B79 Pages: 51-62

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] A survey on the lace expansion for the nearest-neighbor models on the BCC lattice2019

    • Author(s)
      Handa Satoshi、Kamijima Yoshinori、Sakai Akira
    • Journal Title

      Taiwanese Journal of Mathematics

      Volume: - Issue: 3 Pages: 723-784

    • DOI

      10.11650/tjm/190904

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees2019

    • Author(s)
      Sakai Akira、Slade Gordon
    • Journal Title

      Electronic Journal of Probability

      Volume: 24 Issue: none

    • DOI

      10.1214/19-ejp327

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Critical two-point function for long-range models with power-law couplings: The marginal case for d≧dc2019

    • Author(s)
      Chen Lung-Chi、Sakai Akira
    • Journal Title

      Communications in Mathematical Physics

      Volume: 372 Issue: 2 Pages: 543-572

    • DOI

      10.1007/s00220-019-03385-9

    • NAID

      120006937027

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Stability of the critical behavior of the Ising model against quantum perturbation2023

    • Author(s)
      Akira Sakai
    • Organizer
      Workshop on Probabilistic Methods in Statistical Mechanics of Random Media and Random Fields 2023
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Mixing time and simulated annealing for the stochastic cellular automata, and beyond2022

    • Author(s)
      Akira Sakai
    • Organizer
      Workshop on Probabilistic Methods in Statistical Mechanics of Random Media and Random Fields 2022
    • Related Report
      2022 Annual Research Report 2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Spread-out limit of the critical points for various statistical-mechanics models2022

    • Author(s)
      Akira Sakai
    • Organizer
      Probability and Analysis on Random Structures and Related Topics
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Stability of the phase transition and critical behavior of the Ising model against quantum perturbation2022

    • Author(s)
      坂井 哲
    • Organizer
      確率論シンポジウム
    • Related Report
      2022 Annual Research Report
  • [Presentation] Stability of the critical behavior of the Ising model against quantum perturbation2022

    • Author(s)
      坂井 哲
    • Organizer
      東京確率論セミナー
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Mixing time and simulated annealing for the stochastic cellular automata, and beyond2021

    • Author(s)
      Akira Sakai
    • Organizer
      確率論シンポジウム
    • Related Report
      2021 Research-status Report
  • [Presentation] Mixing time and simulated annealing for the stochastic cellular automata2020

    • Author(s)
      Akira Sakai
    • Organizer
      International Workshop on Microstructure-based Global Analysis and Its Related Topics
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Finding optimal solutions by stochastic cellular automata2019

    • Author(s)
      Sakai Akira
    • Organizer
      Workshop on Probabilistic Methods in Statistical Mechanics of Random Media and Random Fields
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical two-point function for long-range models with power-law couplings: The marginal case for d≧dc2019

    • Author(s)
      Sakai Akira
    • Organizer
      The 12th MSJ-SI “Stochastic Analysis, Random Fields and Integrable Probability”
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Finding optimal solutions by stochastic cellular automata2019

    • Author(s)
      Sakai Akira
    • Organizer
      7th Wellington Workshop in Probability and Mathematical Statistics
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical two-point function for long-range models with power-law couplings: The marginal case for d≧dc2019

    • Author(s)
      Akira Sakai
    • Organizer
      Auckland Seminar in Statistics
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Finding optimal solutions by stochastic cellular automata2019

    • Author(s)
      Akira Sakai
    • Organizer
      AIMaP 1-day Workshop
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Hyperscaling for oriented percolation in 1+1 space-time dimensions2018

    • Author(s)
      Akira Sakai
    • Organizer
      Rikkyo Math Phys Seminar
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Critical two-point function for long-range models with power-law couplings: The marginal case for d≧dc2018

    • Author(s)
      Akira Sakai
    • Organizer
      2018 Spring Probability Workshop
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical two-point function for long-range models with power-law couplings: The marginal case for d≧dc2018

    • Author(s)
      Akira Sakai
    • Organizer
      High-dimensional Critical Phenomena in Random Environments
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical two-point function for long-range models with power-law couplings: The marginal case for d≧dc2018

    • Author(s)
      Akira Sakai
    • Organizer
      17th International Symposium “Stochastic Analysis on Large-scale Interacting Systems”
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] Sojourns in Probability Theory and Statistical Physics - I2019

    • Author(s)
      Sidoravicius Vladas (Ed.)
    • Total Pages
      338
    • Publisher
      Springer Singapore
    • ISBN
      9789811502941
    • Related Report
      2019 Research-status Report
  • [Remarks] 個人ホームページ

    • URL

      https://www.math.sci.hokudai.ac.jp/~sakai/

    • Related Report
      2022 Annual Research Report
  • [Remarks] ResearchGate

    • URL

      https://www.researchgate.net/profile/Akira-Sakai

    • Related Report
      2022 Annual Research Report
  • [Remarks] researchmap

    • URL

      https://researchmap.jp/akira_sakai/?lang=ja

    • Related Report
      2022 Annual Research Report
  • [Remarks] Google Scholar

    • URL

      https://scholar.google.com/citations?user=Q7ftcFEAAAAJ&hl=en

    • Related Report
      2022 Annual Research Report
  • [Funded Workshop] The RIMS Workshop “Rigorous Statistical Mechanics and Related Topics IV”2022

    • Related Report
      2022 Annual Research Report
  • [Funded Workshop] The RIMS Workshop “Rigorous Statistical Mechanics and Related Topics II”2020

    • Related Report
      2020 Research-status Report
  • [Funded Workshop] The RIMS Workshop “Rigorous Statistical Mechanics and Related Topics”2019

    • Related Report
      2019 Research-status Report

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Published: 2018-04-23   Modified: 2024-01-30  

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