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Elkies, Noam D.; Kuperberg, Greg; Larsen, Michael; Propp, James

doi:10.1023/a:1022483817303

Cited by 132 publications

(55 citation statements)

References 20 publications

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“…This general method was introduced in the proofs of Kuperberg [31,32], with certain steps in the unrestricted ASM and VSASM cases being based on previously-known results. For example, a bijection between ASM(n) and configurations of the six-vertex model on an n × n square grid with domain-wall boundary conditions had been observed by Elkies, Kuperberg, Larsen and Propp [24,Sec. 7], and (using different terminology) by Robbins and Rumsey [44, pp.…”

Section: 1mentioning

confidence: 86%

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: 1mentioning

confidence: 86%

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Behrend

Fischer

Konvalinka

2017

Advances in Mathematics

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“…These tilings have been an object of interest for mathematicians, see in particular [12], [18]. The "arctic circle theorem" [11] shows that as the size of the system grows large, the domino configurations become frozen outside the circle inscribed inside the diamond, and remain disordered but still heterogeneous [12] (i.e.…”

Section: Tilings Of the Aztec Diamondmentioning

confidence: 99%

Thermodynamic limit of the six-vertex model with domain wall boundary conditions

Korepin

Zinn-Justin

2000

J. Phys. A: Math. Gen.

154

231

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We address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model. Here we compare the bulk free energy for periodic and domain wall boundary conditions. Using a determinant representation for the partition function with domain wall boundary conditions, we derive Toda differential equations and solve them asymptotically in order to extract the bulk free energy. We find that it is different and bears no simple relation with the free energy for periodic boundary conditions. The six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics (alternating sign matrices). This implies new results for the weighted counting for large size alternating sign matrices. Finally we comment on the interpretation of our results, in particular in connection with domino tilings (dimers on a square lattice). 04/2000

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“…Recently, Propp and coworkers [14] furnished the above algebraic reduction with an elegant graphical interpretation for the case of bipartite planar lattices [15]. The partition function Z L on a lattice of linear size L is re-…”

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confidence: 99%

Thermodynamics of Mesoscopic Vortex Systems in1+1Dimensions

1999

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The thermodynamics of a disordered planar vortex array is studied numerically using a new polynomial algorithm which circumvents slow glassy dynamics. Close to the glass transition, the anomalous vortex displacement is found to agree well with the prediction of the renormalization-group theory. Interesting behaviors such as the universal statistics of magnetic susceptibility variations are observed in both the dense and dilute regimes of this mesoscopic vortex system.The behavior of vortices in dirty type-II superconductors has been a subject of intense studies in the last decade [1]. Aside from the obvious technological significance of vortex pinning, understanding the physics of such interacting many-body systems in the presence of quenched disorder is a central theme of modern condensed matter physics. Similarities between the randomly-pinned vortex system and the more familiar mesoscopic electronic systems [2] are highlighted by a recent experimental study of a planar vortex array threaded through a thin crystal of 2H-NbSe 2 by Bolle et al. [3]. Interesting behaviors, including the sampledependent magnetic responses known as "finger prints", have been observed for such a mesoscopic vortex system. The disordered planar vortex array is well studied theoretically [4][5][6][7][8][9]. It is one of the few disorder-dominated systems for which quantitative predictions can be made, including a finite-temperature "vortex glass" phase [5] characterized by anomalous vortex displacements [4], and universal variation of magnetic susceptibility [9]. However, until the work of Bolle et al., there were hardly any experimental studies of this system, with difficulties stemming partly from the weak magnetic signals in such 2d systems. Also, numerical simulations have been limited by the slow glassy dynamics [10], although the availability of special optimization algorithms did lead to the elucidation of the zero-temperature problem in recent years [11]. In this letter, we describe numerical studies of the thermodynamics of the vortex glass via a mapping to a discrete dimer model with quenched disorder. A new polynomial algorithm for the dimer problem circumvents the glassy dynamics and enables us to study large systems at finite temperatures. Our results obtained in the dilute (single-flux-pinning) regime compare well with the experiment by Bolle et al. [3], while those obtained in the collective-pinning regime strongly support the renormalization-group theory of the vortex glass, including its prediction of universal susceptibility variation [9].The Model: The dimer model consists of all complete dimer coverings {D} on a square lattice L as illustrated in Fig. 1(a). The partition function iswhere the sum in the exponential is over all dimers of a given covering, and T d is the dimer temperature. Quenched disorder is introduced via random bond energies ǫ ij , chosen independently and uniformly in the interval (− 1 2 , 1 2 ). The dimer model is related to the planar vortex-line array via the well-known mapping to the solid-on-...

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Cited by 132 publications

References 20 publications

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Thermodynamic limit of the six-vertex model with domain wall boundary conditions

Thermodynamics of Mesoscopic Vortex Systems in1+1Dimensions

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