Arctic Curves In Path Models from The Tangent Method

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 vario...

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Bibliographic Details
Published in:Journal of Physics A: Mathematical and Theoretical
Main Authors: Di Francesco, Philippe, Lapa, Matthew, F.
Other Authors: Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Direction de Recherche Fondamentale (CEA) (DRF (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Department of Mathematics Urbana, University of Illinois at Urbana-Champaign Urbana (UIUC), 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://cea.hal.science/cea-01692535
https://cea.hal.science/cea-01692535/document
https://cea.hal.science/cea-01692535/file/1711.03182.pdf
https://doi.org/10.1088/1751-8121/aab3c0
Description
Summary: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.

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