Tetrahedron equation and quantum integrable systems

• Project/Area Number: 15K04892
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: The University of Tokyo
• Principal Investigator: Kuniba Atsuo 東京大学, 大学院総合文化研究科, 教授 (70211886)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 可積分系 / マルコフ過程 / 量子群 / ヤン・バクスター方程式 / 四面体方程式 / 行列積 / 確率過程 / 非対称排他過程 / 非対称ゼロレンジ過程 / 組合せR / 結晶基底

Outline of Final Research Achievements

An integrable model in Markov processes in non-equilibrium statistical mechanics was constructed. It is formulated as a stochastic dynamics of n kinds of particles on one-dimensional lattice obeying asymmetric transition rules. It includes a majority of such models which have been studied so far. The relation to the tetrahedron equation and the theory of crystal base of quantum groups was clarified and a matrix product formula was derived for the stationary states. They revealed the representation theoretical origin of the known algorithm to calculate stationary probabilities.
For n=2, density and current profile of the particles were derived and the physical aspects of the model was clarified both qualitatively and quantitatively.

Research Products

• [Int'l Joint Research] Australian National University(Australia)
• [Int'l Joint Research] Universite' Paris Saclay(フランス)
• [Int'l Joint Research]
• [Int'l Joint Research] オーストラリア国立大学(オーストラリア)
• [Journal Article] Matrix product formula for U_q(A_2^(1))-zero range process (2017)
  • Author(s): Atsuo Kuniba, Masato Okado
  • Journal Title: J. Phys. A. Math. Theor. Volume: 50 Issue: 4 Pages: 044001
  • DOI: 10.1088/1751-8121/50/4/044001
  • NAID: 120006539748
  • URL: https://ocu-omu.repo.nii.ac.jp/records/2020026
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Integrable structure of multispecies zero range process (2017)
  • Author(s): A. Kuniba, M. Okado and S. Watanabe
  • Journal Title: SIGMA Volume: 13 Pages: 044-044
  • DOI: 10.3842/sigma.2017.044
  • Peer Reviewed / Open Access
• [Journal Article] Density and current profiles in U_q(A^{(1)}_2) zero range process (2017)
  • Author(s): A. Kuniba, V. V. Mangazeev
  • Journal Title: Nucl. Phys. B Volume: 922 Pages: 148-185
  • DOI: 10.1016/j.nuclphysb.2017.07.001
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A q-boson representation of Zamolodchikov-Faddeev algebra for stochastic R matrix of U_q(A_n^(1)) (2016)
  • Author(s): Kuniba A., Okado M.
  • Journal Title: Lett. Math. Phys. Volume: 107 Issue: 6 Pages: 1111-1130
  • DOI: 10.1007/s11005-016-0934-7
  • NAID: 120006490442
  • URL: https://ocu-omu.repo.nii.ac.jp/records/2020085
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Multispecies totally asymmetric zero range process: I. Multiline process and combinatorial R (2016)
  • Author(s): Atsuo Kuniba, Shouya Maruyama, Masato Okado
  • Journal Title: J. Integrable Syst. Volume: 1 Issue: 1 Pages: 1-30
  • DOI: 10.1093/integr/xyw002
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Multispecies totally asymmetric zero range process:II. Hat relation and tetrahedron equation (2016)
  • Author(s): Atsuo Kuniba, Shouya Maruyama, Masato Okado
  • Journal Title: J. Integrable Syst. Volume: 1 Issue: 1 Pages: 1-20
  • DOI: 10.1093/integr/xyw008
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Inhomogeneous Generalization of a Multispecies Totally Asymmetric Zero Range Process (2016)
  • Author(s): Atsuo Kuniba, Shouya Maruyama, Masato Okado
  • Journal Title: J. Stat. Phys. Volume: 164 Issue: 4 Pages: 952-968
  • DOI: 10.1007/s10955-016-1555-3
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Stochastic R matrix for Uq(A(1)n) (2016)
  • Author(s): Kuniba A., Mangazeev, Maruyama S, Okado M
  • Journal Title: Nucl. Phys. Volume: 913 Pages: 248-277
  • DOI: 10.1016/j.nuclphysb.2016.09.016
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] COMBINATORIAL YANG-BAXTER MAPS ARISING FROM THE TETRAHEDRON EQUATION (2016)
  • Author(s): Kuniba A.
  • Journal Title: Theoretical and Mathematical Physics Volume: 189 Issue: 1 Pages: 1472-1485
  • DOI: 10.1134/s004057791610007x
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Multispecies TASEP and the tetrahedron equation (2016)
  • Author(s): Kuniba A., Maruyama S., Okado M.
  • Journal Title: J. Phys. A: Math. Theor. Volume: 49 Issue: 11 Pages: 114001-114001
  • DOI: 10.1088/1751-8113/49/11/114001
  • Peer Reviewed / Open Access
• [Journal Article] Tetrahedron equation and generalized quantum groups (2015)
  • Author(s): Kuniba A., Okado M., Sergeev S.
  • Journal Title: J. Phys. A: Math. Theor. Volume: 48 Issue: 30 Pages: 304001-304001
  • DOI: 10.1088/1751-8113/48/30/304001
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Multispecies TASEP and combinatorial R (2015)
  • Author(s): Kuniba A., Maruyama S., Okado M.
  • Journal Title: J. Phys. A: Math. Theor. Volume: 48 Issue: 34 Pages: 34FT02-34FT02
  • DOI: 10.1088/1751-8113/48/34/34ft02
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Matrix Products at Matrix Program (2018)
  • Author(s): 国場敦夫
  • Organizer: Non-equilibrium systems and special functions, University of Melbourne
  • Int'l Joint Research / Invited
• [Presentation] Density and current profiles in Uq(A(1)2) zero range process (2017)
  • Author(s): 国場敦夫
  • Organizer: Integrability in low-dimensional quantum systems, University of Melbourne
  • Int'l Joint Research / Invited
• [Presentation] Tetrahedron equation, 3D reflection equation and generalized quantum groups (2017)
  • Author(s): 国場敦夫
  • Organizer: Combinatorics, statistical mechanics and conformal field theory, University of Melbourne
  • Int'l Joint Research / Invited
• [Presentation] Tetrahedron equation, 3D reflection equation and generalized quantum groups (2017)
  • Author(s): 国場敦夫
  • Organizer: New development in Teichmuller space theory, Okinawa Institute of Science and Technology
  • Int'l Joint Research / Invited
• [Presentation] Matrix products in integrable probability (2017)
  • Author(s): 国場敦夫
  • Organizer: 日本数学会
  • Place of Presentation: 首都大学東京南大沢キャンパス(東京都八王子市）
  • Invited
• [Presentation] 一次元排他過程と四面体方程式 (2016)
  • Author(s): 丸山翔也
  • Organizer: 日本物理学会
  • Place of Presentation: 東北学院大学 (宮城県仙台市)
  • Year and Date: 2016-03-21
• [Presentation] 多状態TAZRP (2016)
  • Author(s): 尾角正人
  • Organizer: 日本数学会
  • Place of Presentation: 筑波大学 （茨城県つくば市）
  • Year and Date: 2016-03-19
• [Presentation] Topics from solvable lattice models I,II,III (2016)
  • Author(s): Atsuo Kuniba
  • Organizer: Winter School on Representation Theory 2016
  • Place of Presentation: 東京大学数理科学研究科(東京都目黒区)
  • Year and Date: 2016-01-08
  • Int'l Joint Research / Invited
• [Presentation] Stochastic R matrix for Uq(A(1)n) (2016)
  • Author(s): 国場敦夫
  • Organizer: 日本数学会
  • Place of Presentation: 関西大学千里山キャンパス(大阪府吹田市）
• [Presentation] Tetrahedron equation and matrix product method (2016)
  • Author(s): 国場敦夫
  • Organizer: Recent advances in quantum integrable systems
  • Place of Presentation: ジュネーブ(スイス)
  • Int'l Joint Research / Invited
• [Presentation] n-状態TASEPと量子群の組み合わせR (2015)
  • Author(s): 丸山翔也
  • Organizer: 日本物理学会
  • Place of Presentation: 関西大学（大阪府吹田市)
  • Year and Date: 2015-09-16
• [Presentation] 多状態TASEPと四面体方程式 (2015)
  • Author(s): 国場敦夫
  • Organizer: 日本数学会
  • Place of Presentation: 京都産業大学(京都府京都市）
  • Year and Date: 2015-09-13
• [Presentation] Combinatorial Bethe ansatz and ultradiscrete integrable systems (2015)
  • Author(s): Atsuo Kuniba
  • Organizer: XVIIIth International Congress on Mathematical Physics
  • Place of Presentation: Santiago, Chile
  • Year and Date: 2015-07-30
  • Int'l Joint Research / Invited
• [Presentation] Multispecies TASEP and the tetrahedron equation (2015)
  • Author(s): Atsuo Kuniba
  • Organizer: Baxter 2015, Exactly Solved Models and Beyond
  • Place of Presentation: Palm Cove, Australia
  • Year and Date: 2015-07-23
  • Int'l Joint Research / Invited
• [Presentation] Tetrahdron equation and generalized quantum groups (2015)
  • Author(s): Atsuo Kuniba
  • Organizer: Physics and Mathematics of Nonlinear Phenomena 2015
  • Place of Presentation: Gallipoli, Italy
  • Year and Date: 2015-06-25
  • Int'l Joint Research / Invited
• [Remarks]
  • URL: http://webpark1739.sakura.ne.jp/atsuo/indexLP.html
• [Funded Workshop] Algebraic and Combinatorial Aspects in Integrable Systems (2017)
• [Funded Workshop] Integrable Hierarchies and Beyond (2016)
  • Place of Presentation: 名古屋大学多元数理研究科（愛知県名古屋市）
  • Year and Date: 2016-08-29
• [Funded Workshop] Infinite Analysis 16, New Developments in Integrable Systems (2016)
  • Place of Presentation: 大阪市立大学(大阪府大阪市）
  • Year and Date: 2016-03-24