Tangent method for the arctic curve arising from freezing boundaries

Debin, Bryan; Ruelle, Philippe

doi:10.48550/arxiv.1810.04909

Cited by 3 publications

(4 citation statements)

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“…the liquid phase from new frozen domains directly induced by the boundary conditions (hence the denomination 'freezing boundaries'). Quite recently, this conjecture was proved in all generality by Debin and Ruelle in [DR18] for the q = 1 version of the model. There it was shown how to extend the tangent method to arbitrary freezing boundaries and get these new portions of arctic curve by performing some clever shift below the x-axis of the starting points for those paths originally originating from one of the extremities of the freezing boundary.…”

Section: Portions Induced By Freezing Boundariesmentioning

confidence: 88%

A tangent method derivation of the arctic curve for q-weighted paths with arbitrary starting points

Francesco¹,

Guitter²

2019

J. Phys. A: Math. Theor.

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We use a tangent method approach to obtain the arctic curve in a model of non-intersecting lattice paths within the first quadrant, including a q-dependent weight associated with the area delimited by the paths. Our model is characterized by an arbitrary sequence of starting points along the positive horizontal axis, whose distribution involves an arbitrary piecewise differentiable function. We give an explicit expression for the arctic curve in terms of this arbitrary function and of the parameter q. A particular emphasis is put on the deformation of the arctic curve upon varying q, and on its limiting shapes when q tends to 0 or infinity. Our analytic results are illustrated by a number of detailed examples.

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Section: Portions Induced By Freezing Boundariesmentioning

confidence: 88%

A tangent method derivation of the arctic curve for q-weighted paths with arbitrary starting points

Francesco¹,

Guitter²

2019

J. Phys. A: Math. Theor.

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“…The tangent method, as it is known, was the result of a surprising geometric observation: these authors noticed that the arctic curve computed in [4] for the six-vertex model is the caustic of a family of straight lines determined by a one-point boundary observable. It was then elevated to a universal geometric principle and successfully checked in a number of situations where it indeed reproduced known results [12][13][14][15][16][17]. It has been subsequently used to make predictions in cases the analytic shape of the arctic curve was not known, see [12,14,[17][18][19][20].…”

Section: J Stat Mech (2019) 113107mentioning

confidence: 94%

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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“…In the analogous problem on the square grid, there are three possible types of frozen region: empty (no paths), horizontal paths, and vertical paths. The parametrization of these freezing boundaries was worked out in [11]. In our case, the lecture hall tableaux do not appear to develop frozen regions of horizontal paths.…”

Section: Last Dual Pathmentioning

confidence: 96%

“…Other rigorous methods use for example the machinery of cluster integrable systems of dimers [16,26,28]. Recently several papers use the recent method of Colomo and Sportiello [7,8] called the tangent method to compute (non rigorously) the arctic curves [12,13,14,15,11]. A very recent preprint of Aggarwal builds a method to make this heuristic rigorous in the case of the 6-vertex model [1].…”

Section: Introductionmentioning

confidence: 99%

Arctic curves phenomena for bounded lecture hall Tableaux

Corteel¹,

Keating²,

Nicoletti³

2019

Preprint

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Recently the first author and Jang Soo Kim introduced lecture hall tableaux in their study of multivariate little q-Jacobi polynomials. They then enumerated bounded lecture hall tableaux and showed that their enumeration is closely related to standard and semistandard Young tableaux. In this paper we study the asymptotic behavior of these bounded tableaux thanks to two other combinatorial models: non intersecting paths on a graph whose faces are squares and pentagons and dimer models on a lattice whose faces are hexagons and octogons. We use the tangent method to investigate the arctic curve in the model of nonintersecting lattice paths with fixed starting points and ending points distributibuted according to some arbitrary piecewise differentiable function. We then study the dimer model and use some ansatz to guess the asymptotics of the inverse of the Kasteleyn matrix confirm the arctic curve computed with the tangent method for two examples.

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Tangent method for the arctic curve arising from freezing boundaries

Cited by 3 publications

References 0 publications

A tangent method derivation of the arctic curve for q-weighted paths with arbitrary starting points

A tangent method derivation of the arctic curve for q-weighted paths with arbitrary starting points

Concavity analysis of the tangent method

Arctic curves phenomena for bounded lecture hall Tableaux

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