Concavity analysis of the tangent method

Abstract

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.

Arctic boundaries of the ice model on three-bundle domains

“…The tangent method was implemented in [20] to predict the arctic boundary of the domainwall six-vertex model; the result matched earlier predictions from [15]. It was also used later to heuristically derive the arctic boundaries for the six-vertex model on other domains [20,19] by Colomo-Sportiello and Colomo-Pronko-Sportiello; for vertically symmetric alternating sign matrices [32] by Di Francesco-Lapa; various classes of non-intersecting path models [27,28,29,30,31,32] by Debin-Granet-Ruelle, Debin-Ruelle, Di Francesco-Guitter, and Di Francisco-Lapa; for twentyvertex models by Debin-Di Francesco-Guitter [26]; and for random lecture hall tableaux by Corteel-Keating-Nicoletti [22].…”

Factorization in the multirefined tangent method

“…In case the weight of a path is the product of the weights of its elementary steps, the function L is computable in terms of the steps in S and their weights, see the Appendix A for a brief account. Under the same assumptions, it was proved that the function L is strictly concave [DGR19]. In the general case, the concavity of L is easy to establish, but we are not aware of a general result ensuring its strict concavity, though the property has been seen to hold in a number of examples, among which the generic 6-vertex model (see Appendix A).…”

“…The factorization we want to discuss is suggested by the analysis we have presented in [DGR19] to prove the tangency property assumed in the tangent method. It has been sketched, and written explicitly in special cases, by Sportiello [Sp15,Sp19] under the names of entropic tangent method and 2-refined tangent method, described as alternative and tentatively more rigorous ways to compute the shape of arctic curves.…”

“…But it clearly postulates a separation between the outermost path and the other paths. It served as a basis in [DGR19] to address the second assumption, which was proved to hold in a fairly general setting. The analysis relied on this separation in the following way.…”

“…By using the strict concavity of L, it has been shown [DGR19] that the maximum f * within the set of continuous and piecewise C 1 functions is indeed unique, is of class C 1 and corresponds to the path that minimizes the distance between the starting and the ending points while remaining in D. As a corollary, if the domain D allows it, the function f * is the straight line from P i to P f . If not (when the straight line would cross the boundary of D), the function f * (x) is composed of straight segments in the interior of D and portions of the boundary of D, namely pieces of the arctic curve.…”