A tangent method derivation of the arctic curve for $q$-weighted paths with arbitrary starting points

• Published in: Journal of Physics A: Mathematical and Theoretical
• Main Authors: Di Francesco, Philippe, Guitter, Emmanuel
• Other Authors: Department of Mathematics, Illinois State University, Illinois State University, 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), Université Paris-Saclay, ANR-14-CE25-0014,GRAAL,GRaphes et Arbres ALéatoires(2014)
• Format: Article in Journal/Newspaper
• Language: English
• Published: HAL CCSD 2019
• Subjects: non-intersecting lattice paths; continuum limit; arctic curve; [PHYS]Physics [physics]; Arctic
• Online Access: https://hal-cea.archives-ouvertes.fr/cea-02932285; https://hal-cea.archives-ouvertes.fr/cea-02932285/document; https://hal-cea.archives-ouvertes.fr/cea-02932285/file/1810.07936v1.pdf; https://doi.org/10.1088/1751-8121/ab03ff

International audience We use a tangent method approach to obtain the arctic curve in a model of non-intersecting lattice paths within the first quadrant, including a $q$-dependent weight associated with the area delimited by the paths. Our model is characterized by an arbitrary sequence of starting points along the positive horizontal axis, whose distribution involves an arbitrary piecewise differentiable function. We give an explicit expression for the arctic curve in terms of this arbitrary function and of the parameter $q$. A particular emphasis is put on the deformation of the arctic curve upon varying $q$, and on its limiting shapes when $q$ tends to $0$ or infinity. Our analytic results are illustrated by a number of detailed examples.