Fully Packed Loop Models on Finite Geometries

Gier, Jan de

doi:10.1007/978-1-4020-9927-4_13

Cited by 3 publications

(5 citation statements)

References 82 publications

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“…Then, for each (n − 1) × (n − 1) matrix Ā which can be obtained from this process (of which there are 2 S(A) cases), construct an n × n matrix A, where the strictly upper and lower triangular parts of A are given by the upper and lower triangular parts, respectively, of Ā, and the main diagonal of A is obtained by requiring that the sum of entries in each row (or column) is 1. It can easily be checked that each such A is an element of DSASM(n), and that each element of DSASM(n) is obtained exactly once by taking all A ∈ DSASM(n − 1), thereby proving (24).…”

Section: 4mentioning

confidence: 91%

“…• In addition to the formula (43) proved in this paper for the ratio between the numbers of odd-order DASASMs with central entry −1 and 1, analogous formulae have been proved for HTSASMs by Razumov • Various results and conjectures are known for the refined enumeration of several classes of ASMs with respect to so-called link patterns of associated fully packed loop configurations. For further information, see for example Cantini and Sportiello [15,16], and de Gier [21].…”

Section: 1mentioning

confidence: 99%

“…• Various results and conjectures are known for the refined enumeration of several classes of ASMs with respect to so-called link patterns of associated fully packed loop configurations. For further information, see for example Cantini and Sportiello [15,16], and de Gier [21].…”

Section: (A5) and (A6)]mentioning

confidence: 99%

“…The associated partition function (17) consists of a sum, over all such configurations, of products of certain parameterized bulk and boundary weights, as given in Table 2, with these weights satisfying the Yang-Baxter and reflection equations, (47) and (48). Straight sums over all configurations, or over all configurations which correspond to DASASMs with a fixed central entry, are obtained for certain specializations, (21), (22) and (24), of the parameters in the partition function. By identifying particular properties which uniquely characterize the partition function (including symmetry with respect to certain parameters, and reduction to a lower order partition function at certain values of some of the parameters), it is shown in Section 3.5 that the partition function can be expressed as a sum of two determinantal terms, as given in Theorem 1.…”

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confidence: 99%

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Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Behrend

Fischer

Konvalinka

2017

Advances in Mathematics

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Abstract. We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of (2n+1)×(2n+1) DASASMs is ∏ n i=0 (3i)! (n+i)! , and a conjecture of Stroganov from 2008 that the ratio between the numbers of (2n + 1) × (2n + 1) DASASMs with central entry −1 and 1 is n (n + 1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.

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Section: 4mentioning

confidence: 91%

Section: 1mentioning

confidence: 99%

Section: (A5) and (A6)]mentioning

confidence: 99%

mentioning

confidence: 99%

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Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Behrend

Fischer

Konvalinka

2017

Advances in Mathematics

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“…Our construction included a proof of the existence of fully packed loops on AB, in which every vertex is visited by a loop, but these need no longer be the same loop [86]. The FPL model is important for understanding the ground states of frustrated magnetic materials such as the spin ices [87].…”

Section: Discussionmentioning

confidence: 99%

Hamiltonian cycles on Ammann-Beenker Tilings

Singh¹,

Lloyd²,

Flicker³

2023

Preprint

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We provide a simple algorithm for constructing Hamiltonian graph cycles (visiting every vertex exactly once) on the set of aperiodic two-dimensional Ammann-Beenker (AB) tilings. Using this result, and the discrete scale symmetry of AB tilings, we find exact solutions to a range of other problems including the minimum dominating set problem, the domatic number problem, the longest path problem, and the induced path problem. Of direct relevance to physics, we solve the equalweight travelling salesperson problem, providing for example the most efficient route a scanning tunneling microscope tip could take to image the atoms of physical quasicrystals, and the 3-colouring problem, giving ground states for the 3-state Potts model of magnetic interactions. We discuss the adsorption of chain molecules onto quasicrystal surfaces, with possible applications to catalysis.

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Stochastic six-vertex model

Borodin¹,

Corwin²,

Gorin³

2016

Duke Math. J.

119

210

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Abstract. We study the asymmetric six-vertex model in the quadrant with parameters on the stochastic line. We show that the random height function of the model converges to an explicit deterministic limit shape as the mesh size tends to 0. We further prove that the one-point fluctuations around the limit shape are asymptotically governed by the GUE Tracy-Widom distribution. We also explain an equivalent formulation of our model as an interacting particle system, which can be viewed as a discrete time generalization of ASEP started from the step initial condition. Our results confirm an earlier prediction of Gwa and Spohn (1992) that this system belongs to the KPZ universality class.

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Fully Packed Loop Models on Finite Geometries

Cited by 3 publications

References 82 publications

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Hamiltonian cycles on Ammann-Beenker Tilings

Stochastic six-vertex model

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