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Register a new userThe Rotor Model and Combinatorics
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Authors
M.T. Batchelor
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We examine the groundstate wavefunction of the rotor model for different boundary conditions. Three conjectures are made on the appearance of numbers enumerating alternating sign matrices. In addition to those occurring in the O($n=1$) model we find the number $A_{rm V}(2m+1;3)$, which 3-enumerates vertically symmetric alternating sign matrices.
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Explicit algebraic area enumeration formulae are derived for various lattice walks generalizing the canonical square lattice walk, and in particular for the triangular lattice chiral walk recently introduced by the authors. A key element in the enumeration is the derivation of some remarkable identities involving trigonometric sums --which are also important building blocks of non trivial quantum models such as the Hofstadter model-- and their explicit rewriting in terms of multiple binomial sums. An intriguing connection is also made with number theory and some classes of Apery-like numbers, the cousins of the Apery numbers which play a central role in irrationality considerations for {zeta}(2) and {zeta}(3).
In this note, we give a closed formula for the partition function of the dimer model living on a (2 x n) strip of squares or hexagons on the torus for arbitrary even n. The result is derived in two ways, by using a Potts model like description for the dimers, and via a recursion relation that was obtained from a map to a 1D monomer-dimer system. The problem of finding the number of perfect matchings can also be translated to the problem of finding a minmal feedback arc set on the dual graph.
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