The Loop-Weight Changing Operator in the Completely Packed Loop Model

  • Bernard NienhuisEmail author
  • Kayed Al Qasimi
Part of the MATRIX Book Series book series (MXBS, volume 2)


Loop models are statistical ensembles of closed paths on a lattice. The most well-known among them has a variety of names such as the dense O(n) loop model, the Temperley-Lieb (TL) model. This note concerns the model in which the weight of the loop n = 1, and a local operator which changes the weight of all the loops that surround the position of the operator to some other value. A conjecture of the expectation value of the one-point function of this operator was formulated 15 years ago. In this note we sketch the proof.


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We thank Christian Hagendorf for his important contribution to this result, and the MATRIx organization for the excellent opportunity for scientific exchange.


  1. 1.
    Baxter, R.J., Kelland, S.B., Wu, F.Y.: Equivalence of Potts model or Whitney polynomial with an ice-type model. J. Phys. A 9 397 (1976)CrossRefGoogle Scholar
  2. 2.
    Razumov, A.V., Stroganov, Y.G.: Spin chains and combinatorics. J. Phys. A 34, 3185 (2001). arXiv:cond-mat/0012141Google Scholar
  3. 3.
    Kuperberg, G.: Symmetry classes of alternating-sign matrices under one roof. Ann. Math. 156, 835 (2002). arXiv:math/0008184Google Scholar
  4. 4.
    Batchelor, M.T., de Gier, J., Nienhuis, B.: The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions. J. Phys. A 34, L265 (2001). arXiv:cond-mat/0101385Google Scholar
  5. 5.
    Razumov, A.V., Stroganov, Y.G.: Combinatorial nature of ground state vector of O(1) loop model. Theor. Math. Phys. 138, 333 (2004); Teor.Mat.Fiz. 138 (2004) 395; arXiv:math/0104216Google Scholar
  6. 6.
    Cantini, L., Sportiello, A.: Proof of the Razumov-Stroganov conjecture. J. Combin. Theor. A 118, 1549 (2011). arXiv:1003.337Google Scholar
  7. 7.
    Mitra, S., Nienhuis, B.: Exact conjectured expressions for correlations in the dense O(1) loop model on cylinders. J. Stat. Mech. P10006 (2004). arXiv:cond-mat/0407578Google Scholar
  8. 8.
    Hagendorf, C., Morin-Duchesne, A.: Symmetry classes of alternating sign matrices in the nineteen-vertex model. J. Stat. Mech. 053111 (2016). arXiv:1601.01859Google Scholar
  9. 9.
    Di Francesco, P., Zinn-Justin, P.: Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule. Electron. J. Combin. 12, R6 (2005). arXiv:math-ph/0410061Google Scholar
  10. 10.
    Knizhnik, V.G., Zamolodchikov, A.B.: Current-algebra and Wess-Zumino model in 2 dimensions. Nucl. Phys. B 247, 83 (1984)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Di Francesco, P., Zinn-Justin, P., Zuber, J.-B.: Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain. J. Stat. Mech. P08011 (2006). arXiv:math-ph/0603009Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of PhysicsAmsterdamThe Netherlands
  2. 2.Korteweg de Vries InstituteAmsterdamThe Netherlands

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