Arctic Curves In Path Models from The Tangent Method

• Published in: Journal of Physics A: Mathematical and Theoretical
• Main Authors: Di Francesco, Philippe, Lapa, Matthew
• Other Authors: Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics Urbana, University of Illinois at Urbana-Champaign Urbana, University of Illinois System-University of Illinois System
• Format: Article in Journal/Newspaper
• Language: English
• Published: HAL CCSD 2018
• Subjects: [PHYS]Physics [physics]; Arctic
• Online Access: https://hal-cea.archives-ouvertes.fr/cea-01692535; https://hal-cea.archives-ouvertes.fr/cea-01692535/document; https://hal-cea.archives-ouvertes.fr/cea-01692535/file/1711.03182.pdf; https://doi.org/10.1088/1751-8121/aab3c0

International audience Recently, Colomo and Sportiello introduced a powerful method, known as the $Tangent\ Method$, for computing the arctic curve in statistical models which have a (non-or weakly-) intersecting lattice path formulation. We apply the Tangent Method to compute arctic curves in various models: the domino tiling of the Aztec diamond for which we recover the celebrated arctic circle; a model of Dyck paths equivalent to the rhombus tiling of a half-hexagon for which we find an arctic half-ellipse; another rhombus tiling model with an arctic parabola; the vertically symmetric alternating sign matrices, where we find the same arctic curve as for unconstrained alternating sign matrices. The latter case involves lattice paths that are non-intersecting but that are allowed to have osculating contact points, for which the Tangent Method was argued to still apply. For each problem we estimate the large size asymptotics of a certain one-point function using LU decomposition of the corresponding Gessel-Viennot matrices, and a reformulation of the result amenable to asymptotic analysis.