Studies on two dimensional critical stochastic process by exactly solvable models and field theory

• Project/Area Number: 24540393
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical physics/Fundamental condensed matter physics
• Research Institution: The University of Tokyo
• Principal Investigator: Sakai Kazumitsu 東京大学, 総合文化研究科, 助教 (10397028)
• Co-Investigator(Kenkyū-buntansha): 茂木 康平 東京海洋大学, 海洋科学技術研究科, 助教 (30583033)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 確率過程 / 臨界現象 / シュラム・レヴナー発展 / 共形場理論 / 可解模型 / 可積分系 / グロタンディーク多項式 / K理論 / Schramm-Loewner発展 / コセット構成 / パラフェルミオン模型 / 対称多項式 / ソリトン / SLE / 非平衡系 / 厳密解 / 確率的レーブナー方程式 / ベーテ仮説

Outline of Final Research Achievements

Random fractals are a geometric structure universally appearing in the critical phenomena. Applying the Schramm-Loewner evolution, we have investigated the structure of random fractals observed in the two dimensional critical phenomena.
Furthermore, some relation between solvable models and geometry has been studied. Namely, we have found that some integrable models are related to the K theory defined on complex manifolds.

Research Products

• [Journal Article] Quantum integrable combinatorics of Schur polynomials (2015)
  • Author(s): K. Motegi, K. Sakai
  • Journal Title: arXiv:1507.06740 Volume: arXiv:1507.06740
  • Acknowledgement Compliant
• [Journal Article] Partition functions of integrable lattice models and combinatorics of symmetric polynomials (2015)
  • Author(s): K. Motegi, K. Sakai, S. Watanabe
  • Journal Title: arXiv:1512.07955 Volume: arXiv:1512.07955
  • Acknowledgement Compliant
• [Journal Article] K-theoretic boson-fermion correspondence and melting crystals (2014)
  • Author(s): K. Motegi and K. Sakai
  • Journal Title: Journal of Physics A: Mathematical and Theoretical Volume: 47 Issue: 44 Pages: 445202-445202
  • DOI: 10.1088/1751-8113/47/44/445202
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Thermal Pure Quantum States of Many-Particle Systems (2014)
  • Author(s): Masahiko. Hyuga, Sho Sugiura, Kazumitsu Sakai, and Akira Shimizu
  • Journal Title: Physical Review B Volume: 90 Issue: 12
  • DOI: 10.1103/physrevb.90.121110
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Nonstandard representations of type C affine Hecke algebra from K-operators (2014)
  • Author(s): K. Motegi
  • Journal Title: arXiv:1409.0216 Volume: arXiv:1409.0216
  • Acknowledgement Compliant
• [Journal Article] Vertex models, TASEP and Grothendieck polynomials (2013)
  • Author(s): K. Motegi and K. Sakai
  • Journal Title: Journal of Physics A: Mathematical and Theoretical Volume: 46 Issue: 35 Pages: 355201-355201
  • DOI: 10.1088/1751-8113/46/35/355201
  • Peer Reviewed
• [Journal Article] K-theoretic boson-fermion correspondence and melting crystals (2013)
  • Author(s): K. Motegi and K. Sakai
  • Journal Title: arXiv:1311.6076 Volume: 1311.6076
• [Journal Article] Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries (2013)
  • Author(s): K. Sakai
  • Journal Title: Nucl. Phys. B Volume: 867 Issue: 2 Pages: 429-447
  • DOI: 10.1016/j.nuclphysb.2012.09.019
  • Peer Reviewed
• [Journal Article] Absence of finite size correction at the combinatorial point of the integrable higher spin XXZ chain (2013)
  • Author(s): K. Motegi
  • Journal Title: arXiv:1301.6311 Volume: arXiv
• [Journal Article] Exact Relaxation Dynamics in the Totally Asymmetric Simple Exclusion Process (2012)
  • Author(s): K.Motegi, K.Sakai, J.Sato
  • Journal Title: Physical Review E Volume: 85 Issue: 4 Pages: 42105-42105
  • DOI: 10.1103/physreve.85.042105
  • Peer Reviewed
• [Journal Article] Long time asymptotics of the totally asymmetric simple exclusion process (2012)
  • Author(s): K. Motegi, K. Sakai and J. Sato
  • Journal Title: J. Phys. A: Math. Theor. Volume: 45 Issue: 46 Pages: 465004-465004
  • DOI: 10.1088/1751-8113/45/46/465004
  • Peer Reviewed
• [Journal Article] On Baxter's Q operator of the higher spin XXZ chain at the Razumov-Stroganov point (2012)
  • Author(s): K. Motegi
  • Journal Title: arXiv:1212.1575 Volume: arXiv
• [Presentation] Multiple SLEs (2016)
  • Author(s): K. Sakai
  • Organizer: Infinite Analysis 16 New Developments in Integrable Systems
  • Place of Presentation: 大阪大学（大阪府、大阪市）
  • Year and Date: 2016-03-24
  • Int'l Joint Research / Invited
• [Presentation] 共形不変性を有するランダムフラクタルの合成則 (2015)
  • Author(s): 堺 和光
  • Organizer: 日本物理学会 2015年秋季大会
  • Place of Presentation: 関西大学（大阪府、吹田市）
  • Year and Date: 2015-09-16
• [Presentation] ２次元成長過程：Schramm-Loewner発展と共形場理論 (2015)
  • Author(s): 堺和光
  • Organizer: 物性理論・統計力学セミナー2015
  • Place of Presentation: ラフォーレ箱根強羅（神奈川県，箱根町）
  • Year and Date: 2015-03-14
• [Presentation] Quantum integrable combinatorics of Schur and Grothendieck polynomials (2015)
  • Author(s): K. Motegi
  • Organizer: International Workshop "Integrable Systems and Representation Theory"
  • Place of Presentation: 東京水産大学（東京）
  • Year and Date: 2015-03-11
  • Invited
• [Presentation] Integrable models and Grothendieck polynomials (2014)
  • Author(s): K. Sakai
  • Organizer: JSPS Japan-Hungary Research Cooperative Program Workshop on “ Integrability in gauge-gravity duality and strong coupling dynamics of gauge theory”
  • Place of Presentation: 京都大学（京都府，京都市）
  • Year and Date: 2014-09-23
  • Invited
• [Presentation] 量子ハイゼンベルク模型における量子的ソリトン (2014)
  • Author(s): 堺和光，古山敦之
  • Organizer: 日本物理学会 2014年秋季大会
  • Place of Presentation: 中部大学（愛知県，名古屋市）
  • Year and Date: 2014-09-09
• [Presentation] グランドカノニカルTPQ状態とカノニカルTPQ状態の比較 (2014)
  • Author(s): 日向理彦，杉浦祥，堺和光，清水明
  • Organizer: 日本物理学会 2014年秋季大会
  • Place of Presentation: 中部大学（愛知県，名古屋市）
  • Year and Date: 2014-09-09
• [Presentation] Vertex models, TASEP and Grothendieck polynomials
  • Author(s): 茂木康平，堺和光
  • Organizer: 日本物理学会 ２０１３年秋季大会
  • Place of Presentation: 徳島大学（徳島県，徳島市）
• [Presentation] XXZ鎖のRazumov-Stroganov点におけるBaxterのQ演算子
  • Author(s): 茂木康平
  • Organizer: 日本物理学会 ２０１３年秋季大会
  • Place of Presentation: 徳島大学（徳島県，徳島市）
• [Presentation] Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
  • Author(s): K. Sakai
  • Organizer: The IUPAP 25th, International Conference on Statistical Physics (STATPHYS 25)
  • Place of Presentation: ソウル国立大学（韓国，ソウル）
• [Presentation] Quantum inverse scattering approach to the totally asymmetric simple exclusion process
  • Author(s): K. Motegi
  • Organizer: The IUPAP 25th, International Conference on Statistical Physics (STATPHYS 25)
  • Place of Presentation: ソウル国立大学（韓国，ソウル）
• [Presentation] Stochastic integrable models and Grothendieck polynomials
  • Author(s): K. Motegi
  • Organizer: A satellite meeting of STATPHYS 25 (2013) and a YITP workshop. Mathematical Statistical Physics
  • Place of Presentation: 京都大学（京都府，京都市）
• [Presentation] Quantum integrable models and Grothendieck polynomials
  • Author(s): K. Motegi
  • Organizer: Infinite Analysis 14: New Perspectives in Integrable Systems and Representation Theory
  • Place of Presentation: 東京大学（東京都，目黒区）
  • Invited
• [Presentation] 量子可積分系とGrothendieck多項式(1)：ボソン・フェルミオン対応
  • Author(s): 茂木康平，堺和光
  • Organizer: 日本物理学会 第69回年次大会
  • Place of Presentation: 東海大学（神奈川県，横浜市）
• [Presentation] 量子可積分系とGrothendieck多項式(2)：溶解結晶模型と平面分割
  • Author(s): 茂木康平，堺和光
  • Organizer: 日本物理学会 第69回年次大会
  • Place of Presentation: 東海大学（神奈川県，横浜市）
• [Presentation] 熱的な量子純粋状態によるFermion系の数値解析
  • Author(s): 日向理彦，杉浦祥，堺和光，清水明
  • Organizer: 日本物理学会 第69回年次大会
  • Place of Presentation: 東海大学（神奈川県，横浜市）
• [Presentation] 可解格子模型とGrothendieck 多項式
  • Author(s): 茂木康平，堺和光
  • Organizer: 日本数学会２０１４年度年会
  • Place of Presentation: 学習院大学（東京，豊島区）
• [Presentation] 多重Stochastic Loewner Evolutionに関して
  • Author(s): 堺和光
  • Organizer: 日本物理学会 2012年秋季大会
  • Place of Presentation: 横浜国立大学（神奈川県，横浜市）
• [Presentation] 周期TASEPの長時間における漸近的振る舞い
  • Author(s): 茂木康平，堺和光，佐藤純
  • Organizer: 日本物理学会 第68回年次大会
  • Place of Presentation: 広島大学（広島県，東広島市）
• [Presentation] 量子逆散乱法による完全非対称単純排他過程の解析
  • Author(s): 茂木康平，堺和光，佐藤純
  • Organizer: 日本数学会 2013年年会
  • Place of Presentation: 京都大学（京都府，京都市）
• [Presentation] Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
  • Author(s): K. Sakai
  • Organizer: Recent Advances in Quantum Integrable Systems (Raqis'12)
  • Place of Presentation: Angers大学（フランス）
• [Presentation] Exact relaxation dynamics in the totally asymmetric simple exclusion process
  • Author(s): K. Motegi
  • Organizer: Recent Advances in Quantum Integrable Systems (Raqis'12)
  • Place of Presentation: Angers大学（フランス）