Concavity analysis of the tangent method

• Published in: Journal of Statistical Mechanics: Theory and Experiment
• Main Authors: Debin, Bryan, Granet, Etienne, Ruelle, Philippe
• Other Authors: UCL - SST/IRMP - Institut de recherche en mathématique et physique
• Format: Article in Journal/Newspaper
• Language: English
• Published: Institute of Physics Publishing Ltd. 2019
• Subjects: solvable lattice models; arctic curves; Arctic
• Online Access: http://hdl.handle.net/2078.1/222707; https://doi.org/10.1088/1742-5468/ab43d6

The tangent method has recently been devised by Colomo and Sportiello (2016 J. Stat. Phys. 164 1488–523) 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.