The Limit Shape of Large Alternating Sign Matrices

Colomo, Filippo; Pronko, A. G.

doi:10.1137/080730639

Cited by 42 publications

(73 citation statements)

References 34 publications

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“…The representation (14) has played a central role in the evaluation of the arctic curve of the model [16,17].…”

Section: Nonlocal Correlation Functionsmentioning

confidence: 99%

“…For the six-vertex model with domain wall boundary conditions [12][13][14], which we consider here, some multiple integral representations are available, for example, for the emptiness formation probability [15]. They have already proved useful in the study of phase separation phenomena in the model, in particular, to obtain the arctic curve (frozen boundary of the limit shape) [16,17].…”

Section: Introductionmentioning

confidence: 99%

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Integral Formulas and Antisymmetrization Relations for the Six-Vertex Model

Cantini¹,

Colomo

Pronko

2019

Ann. Henri Poincaré

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We study the relationship between various integral formulas for nonlocal correlation functions of the six-vertex model with domain wall boundary conditions. Specifically, we show how the known representation for the emptiness formation probability can be derived from that for the so-called row configuration probability. A crucial ingredient in the proof is a relation expressing the result of antisymmetrization of some given function with respect to permutations in two sets of its variables in terms of the Izergin-Korepin partition function. This relation generalizes another one obtained by Tracy and Widom in the context of the asymmetric simple exclusion process. arXiv:1906.07636v1 [math-ph]

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“…The representation (14) has played a central role in the evaluation of the arctic curve of the model [16,17].…”

Section: Nonlocal Correlation Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Integral Formulas and Antisymmetrization Relations for the Six-Vertex Model

Cantini¹,

Colomo

Pronko

2019

Ann. Henri Poincaré

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“…This phenomenon was soon observed to be ubiquitous within the context of highly correlated statistical mechanical systems; see, for instance, [1,2,5,6,7,10,12,13,15,16,17,18,19,20,25,28,29,30,32,33,34,35,42,43,44,45,51,54,57,61]. In particular, Cohn-Kenyon-Propp developed a variational principle [12] that prescribes a law of large numbers for random domino tilings on almost arbitrary domains, which was used effectively by to explicitly determine the arctic boundaries of uniformly random lozenge tilings on polygonal domains.…”

mentioning

confidence: 99%

“…Through a series of works [16,14,17,15] starting around 2008, Colomo-Pronko provided such a prediction. In particular, in [16], they introduced a nonlocal correlation function called the emptiness formation probability and explained how one can derive arctic boundaries from its asymptotic behavior.…”

mentioning

confidence: 99%

Arctic boundaries of the ice model on three-bundle domains

Aggarwal

2019

Invent. math.

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In this paper we consider the six-vertex model at ice point on an arbitrary threebundle domain, which is a generalization of the domain-wall ice model on the square (or, equivalently, of a uniformly random alternating sign matrix). We show that this model exhibits the arctic boundary phenomenon, whose boundary is given by a union of explicit algebraic curves. This was originally predicted by Colomo-Sportiello in 2016 as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries, called the (geometric) tangent method. Our proof uses a probabilistic analysis of non-crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case, which might be of independent interest.

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“…Arctic phenomena have been observed in many other models, including lozenge tilings of hexagons [3], the 6-vertex model with domain wall boundary conditions [4,5], alternating sign matrices [6], lozenge tilings of polygonal domains [7,8] and the 2-periodic Aztec diamond [9]. For dimer models on bipartite planar graphs, rather general results have been obtained in [10,11].…”

Section: Introductionmentioning

confidence: 89%

Concavity analysis of the tangent method

Debin¹,

Granet²,

Ruelle³

2019

J. Stat. Mech.

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The tangent method has recently been devised by Colomo and Sportiello [12] as an efficient way to determine the shape of arctic curves. Largely conjectural, it has been tested successfully in a variety of models. However no proof and no general geometric insight have been given so far, either to show its validity or to allow for an understanding of why the method actually works. In this paper, we propose a universal framework which accounts for the tangency part of the tangent method, whenever a formulation in terms of directed lattice paths is available. Our analysis shows that the key factor responsible for the tangency property is the concavity of the entropy (also called the Lagrangean function) of long random lattice paths. We extend the proof of the tangency to q-deformed paths.

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The Limit Shape of Large Alternating Sign Matrices

Cited by 42 publications

References 34 publications

Integral Formulas and Antisymmetrization Relations for the Six-Vertex Model

Integral Formulas and Antisymmetrization Relations for the Six-Vertex Model

Arctic boundaries of the ice model on three-bundle domains

Concavity analysis of the tangent method

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