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Combinatorial nature of the ground-state vector of the \(\mathrm{O}(1)\) loop model. (English) Zbl 1178.82020
Theor. Math. Phys. 138, No. 3, 333-337 (2004); translation from Teor. Mat. Fiz. 138, No. 3, 395-400 (2004).
Summary: Studying a possible connection between the ground-state vector for some special spin systems and the so-called alternating-sign matrices, we find numerical evidence that the components of the ground-state vector of the \(\mathrm{O}(1)\) loop model coincide with the numbers of the states of the so-called fully packed loop model with fixed pairing patterns. The states of the latter system are in one-to-one correspondence with alternating-sign matrices. This allows advancing the hypothesis that the components of the ground-state vector of the \(\mathrm{O}(1)\) loop model coincide with the cardinalities of the corresponding subsets of the alternating-sign matrices. In a sense, our conjecture generalizes the conjecture of Bosley and Fidkowski, which was refined by Cohn and Propp and proved by B. Wieland [Electron. J. Comb. 7, No.1, Research paper R37, 13 p. (2000; Zbl 0956.05015)].

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05A15 Exact enumeration problems, generating functions
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