Concavity analysis of the tangent method

Abstract

The tangent method has recently been devised by Colomo and Sportiello (arXiv:1605.01388 [math-ph]) 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.

arXiv commentpublished version, 22 pages

References in corpus (7)

• Local statistics for random domino tilings of the Aztec diamond
• The arctic curve of the domain-wall six-vertex model
• Arctic curves of the six-vertex model on generic domains: the Tangent Method
• The arctic curve of the domain-wall six-vertex model in its anti-ferroelectric regime
• The limit shape of large alternating sign matrices
• Arctic curve of the free-fermion six-vertex model in an L-shaped domain
• A tangent method derivation of the arctic curve for q-weighted paths with arbitrary starting points

Cited by in corpus (6)

• Arctic curves of the twenty-vertex model with domain wall boundaries
• Exact time evolution formulae in the XXZ spin chain with domain wall initial state
• Arctic curves of the four-vertex model
• Double tangent method for two-periodic Aztec diamonds
• Factorization in the multirefined tangent method
• Arctic curves of the $6$V model with partial DWBC and double Aztec rectangles