Advancement of the lace expansion and its applications

• Project/Area Number: 21740059
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Hokkaido University
• Principal Investigator: SAKAI Akira 北海道大学, 大学院・理学研究院, 准教授 (50506996)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2011: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2010: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000); Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 確率論 / レース展開 / 高次元臨界現象 / 自己回避歩行 / パーコレーション / コンタクトプロセス / イジング模型

Research Abstract

The lace expansion has been one of the few mathematically rigorous approaches to investigate critical behavior in high dimensions. We have extended this methodology to obtain a universal sharp asymptotic expression of the 2-point functions for long-range self-avoiding walk and long-range oriented percolation which are defined by power-law decaying pair potentials. We have also investigated the finite-range(but sufficiently spread-out) critical contact process and proved that the n-point function under the Brownian scaling converges to the(n-1)-point function for the canonical measure of super-Brownian motion.

Research Products

• [Journal Article] Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation (2011)
  • Author(s): Akira Sakai, Lung-Chi Chen
  • Journal Title: The Annals of Probability Volume: Vol.39 Issue: 2 Pages: 507-548
  • DOI: 10.1214/10-aop557
  • NAID: 120002834721
  • Peer Reviewed
• [Journal Article] Large-time asymptotics of the gyration radius for long-range statistical-mechanical models (2011)
  • Author(s): Akira Sakai
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: Vol.21 Pages: 53-62
  • URL: http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu.html
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation (2011)
  • Author(s): Lung-Chi Chen, Akira Sakai
  • Journal Title: The Annals of Probability Volume: 39 Pages: 507-548
  • NAID: 120002834721
  • Peer Reviewed
• [Journal Article] Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension : The higher-point functions (2010)
  • Author(s): Remco van der Hofstad, Akira Sakai
  • Journal Title: Electronic Journal of Probability Volume: Vol.15 Issue: none Pages: 801-894
  • DOI: 10.1214/ejp.v15-783
  • NAID: 120002548789
  • Peer Reviewed
• [Journal Article] Critical behavior and the limit distribution for long-range oriented percolation. II : Spatial correlation (2009)
  • Author(s): Akira Sakai, Lung-Chi Chen
  • Journal Title: Probability Theory and Related Fields Volume: Vol.145 Issue: 3-4 Pages: 435-458
  • DOI: 10.1007/s00440-008-0174-6
  • NAID: 120002513620
  • Peer Reviewed
• [Journal Article] Critical behavior and the limit distribution for long-range oriented percolation. II : Spatial correlation (2009)
  • Author(s): 坂井哲
  • Journal Title: Probability Theory and Related Fields 145 Pages: 435-458
  • NAID: 120002513620
  • Peer Reviewed
• [Presentation] Asymptotic behavior in Z^d of the critical two-point functions for statistical-mechanical models with power-law decaying potentials (2012)
  • Author(s): Akira Sakai
  • Organizer: VU Probability Seminar
  • Place of Presentation: Vrije Universiteit Amsterdam(オランダ)
  • Year and Date: 2012-03-19
• [Presentation] Asymptotic behavior in Z^d of the critical two-point functions for statistical-mechanical models with power-law decaying potentials (2012)
  • Author(s): Akira Sakai
  • Organizer: VU Probability Seminar
  • Place of Presentation: Vrije Universiteit Amsterdam (Netherlands)(招待講演)
  • Year and Date: 2012-03-19
• [Presentation] Asymptotic behavior in Z^d of the critical two-point functions for statistical-mechanical models with power-law decaying potentials (2012)
  • Author(s): Akira Sakai
  • Organizer: New Zealand Probability Workshop & Australia and New Zealand Applied Probability Workshop
  • Place of Presentation: The University of Auckland(ニュージーランド)
  • Year and Date: 2012-01-24
• [Presentation] Asymptotic behavior in Z^d of the critical two-point functions for statistical-mechanical models with power-law decaying potentials (2012)
  • Author(s): Akira Sakai
  • Organizer: New Zealand rrooaoillty workshop & Austialla and New Zealand Applied Probability Workshop
  • Place of Presentation: University of Auckland (New Zealand)(招待講演)
  • Year and Date: 2012-01-24
• [Presentation] Asymptotic behavior in Z^d of the critical two-point functions for long-range statistical-mechanical models in high dimensions (2011)
  • Author(s): Akira Sakai
  • Organizer: The 7^<th> HU and SNU Symposium on Mathematics "Recent Developments in Mathematical Analysis and Related Fields"
  • Place of Presentation: Seoul National University(韓国)
  • Year and Date: 2011-11-17
• [Presentation] Asymptotic behavior in Z^d of the critical two-point functions for long-range statistical-mechanical models in high dimensions (2011)
  • Author(s): Akira Sakai
  • Organizer: The 7^<th> HU and SNU Symposium on Mathematics "Recent Developments in Mathematical Analysis and Related Fields"
  • Place of Presentation: Seoul National University (South Korea)(招待講演)
  • Year and Date: 2011-11-17
• [Presentation] 臨界現象の厳密な解析を目指して (2011)
  • Author(s): 坂井哲
  • Organizer: 語ろう数理解析
  • Place of Presentation: 北海道大学
  • Year and Date: 2011-09-06
• [Presentation] 臨界現象の厳密な解析を目指して (2011)
  • Author(s): 坂井哲
  • Organizer: 語ろう数理解析
  • Place of Presentation: 北海道大学(札幌市)(招待講演)
  • Year and Date: 2011-09-06
• [Presentation] 臨界現象の数理 (2011)
  • Author(s): 坂井哲
  • Organizer: 第8回城崎新人セミナー
  • Place of Presentation: 城崎市民センター(兵庫県豊岡市)
  • Year and Date: 2011-02-16
• [Presentation] Applications of the lace expansion to theφ^4model (2011)
  • Author(s): Akira Sakai
  • Organizer: Conference on Probability Theory, Statistical Physics, and Applications
  • Place of Presentation: New York University Abu Dhabi Institute(UAE)
  • Year and Date: 2011-01-16
• [Presentation] Application of the lace expansion to the ψ^4 model (2011)
  • Author(s): Akira Sakai
  • Organizer: Conference on Probability Theory, Statistical Physics, and Applications
  • Place of Presentation: New York University Abu Dhabi Institute (UAE)(招待講演)
  • Year and Date: 2011-01-16
• [Presentation] Asymptotic behavior in Z^d of the critical two-point functions for long-range statistical-mechanical models in high dimensions (2010)
  • Author(s): 坂井哲
  • Organizer: 確率論シンポジウム
  • Place of Presentation: 京都大学
  • Year and Date: 2010-12-20
• [Presentation] Asymptotic behavior in Z^d of the critical two-point functions for long-range statistical-mechanical models in high dimensions (2010)
  • Author(s): 坂井哲
  • Organizer: 確率論シンポジウム
  • Place of Presentation: 京都大学(京都府)
  • Year and Date: 2010-12-20
• [Presentation] Lace expansion in the past and future (2010)
  • Author(s): Akira Sakai
  • Organizer: The 34^<th> Conference on Stochastic Processes and their Applications
  • Place of Presentation: 千里ライフセンター(大阪市)
  • Year and Date: 2010-09-07
• [Presentation] Lace expansion in the past and future (2010)
  • Author(s): Akira Sakai
  • Organizer: The 34^<th> Conference on Stochastic Processes and their Applications
  • Place of Presentation: 千里ライフサイエンスセンター(大阪府)(招待講演)
  • Year and Date: 2010-09-07
• [Presentation] The lace expansion for lattice models in high dimensions (2010)
  • Author(s): 坂井哲
  • Organizer: 第11回北東数学解析研究会
  • Place of Presentation: 北海道大学
  • Year and Date: 2010-02-23
• [Presentation] The lace expansion for lattice models in high dimensions (2010)
  • Author(s): 坂井哲
  • Organizer: 第11回北東数学解析研究会
  • Place of Presentation: 北海道大学(札幌市)
  • Year and Date: 2010-02-23
• [Presentation] Asymptotic behavior in Z^d of the critical two-point functions for long-range statistical-mechanical models in high dimensions (2009)
  • Author(s): Akira Sakai
  • Organizer: Above the Critical Dimension
  • Place of Presentation: The Institut Henri Poincare(フランス)
  • Year and Date: 2009-12-09
• [Presentation] Asymptotic behavior in Z^d of the critical two-point functions for long-range statistical-mechanical models in high dimensions (2009)
  • Author(s): 坂井哲
  • Organizer: Above the Critical Dimension
  • Place of Presentation: IHP(Paris, フランス)
  • Year and Date: 2009-12-09
• [Presentation] Asymptotic behavior of the gyration radius for long-range selfavoiding walk and long-range oriented percolation (2009)
  • Author(s): 坂井哲
  • Organizer: 第8回大規模相互作用系の確率解析
  • Place of Presentation: 東京大学
  • Year and Date: 2009-10-08
• [Presentation] Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation (2009)
  • Author(s): 坂井哲
  • Organizer: 第8回大規模相互作用系の確率解析
  • Place of Presentation: 東京大学(東京都)
  • Year and Date: 2009-10-08
• [Presentation] Asymptotic behavior of the gyration radius for long-range selfavoiding walk and long-range oriented percolation (2009)
  • Author(s): 坂井哲
  • Organizer: 繰り込み群の数理科学での応用
  • Place of Presentation: 京都大学
  • Year and Date: 2009-09-10
• [Presentation] Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation (2009)
  • Author(s): 坂井哲
  • Organizer: 繰り込み群の数理科学での応用
  • Place of Presentation: 京都大学(京都市)
  • Year and Date: 2009-09-10
• [Presentation] Asymptotic behavior of the gyration radius for long-range selfavoiding walk and long-range oriented percolation (2009)
  • Author(s): Akira Sakai
  • Organizer: Scaling Limits in Models of Statistical Mechanics
  • Place of Presentation: The Mathematisches Forschungsinstitut Oberwolfach(ドイツ)
  • Year and Date: 2009-08-19
• [Presentation] Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation (2009)
  • Author(s): 坂井哲
  • Organizer: Scaling Limits in Models of Statistical Mechanics
  • Place of Presentation: MFO(Oberwolfach, ドイツ)
  • Year and Date: 2009-08-19
• [Remarks] SPAサテライト研究集会"Universality and Scaling Limits in Probability and Statistical Mechanics"主催, 2010年8月30日～9月3日,北海道大学
• [Remarks]
  • URL: http://www.math.sci.hokudai.ac.jp/~sakai/
• [Remarks]
  • URL: http://www.math.sci.hokudai.ac.jp/~sakai
• [Remarks]
  • URL: http://www.math.sci.hokudai.ac.jp/~sakai/
• [Remarks]
  • URL: http://www.math.sci.hokudai.ac.jp/~sakai/