\(O(1)\) loop model with different boundary conditions and symmetry classes of alternating-sign matrices.

*(English)*Zbl 1178.82044
Theor. Math. Phys. 142, No. 2, 237-243 (2005); translation from Teor. Mat. Fiz. 142, No. 2, 284-292 (2005).

Summary: This work is a continuation of our recent paper where we discussed numerical evidence that the numbers of the states of the fully packed loop model with fixed pairing patterns coincide with the components of the ground state vector of the \(O(1)\) loop model with periodic boundary conditions and an even number of sites. We give two new conjectures related to different boundary conditions: we suggest and numerically verify that the numbers of the half-turn symmetric states of the fully packed loop model with fixed pairing patterns coincide with the components of the ground state vector of the \(O(1)\) loop model with periodic boundary conditions and an odd number of sites and that the corresponding numbers of the vertically symmetric states describe the case of open boundary conditions and an even number of sites.

##### MSC:

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |

##### References:

[1] | I. M. Ternov, V. R. Khalilov, and V. N. Rodionov, Interaction of Charged Particles with a Strong Electromagnetic Field[in Russian], MSU, Moscow (1982). |

[2] | L. D. Landau and E. M. Lifshitz, Classical Field Theory[in Russian] (Course of Theoretical Physics, Vol. 2), Nauka, Moscow (1973); English transl., Pergamon, Oxford (1975). · Zbl 0178.28704 |

[3] | V. V. Ternovskii and A. M. Khapaev, Fund. Prikl. Mat., 8, 547-547 (2002). |

[4] | M. M. Khapaev, Averaging in Stability Theory[in Russian], Nauka, Moscow (1986); [English transl.: Averaging in Stability Theory: A Study of Resonance Multi-Frequency Systems, Kluwer, Dordrecht (1993). · Zbl 0782.34057 |

[5] | N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations[in Russian] (4th ed.), Nauka, Moscow (1974); English transl. prev. ed., Gordon and Breach, New York (1961). |

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