Arctic curves for paths with arbitrary starting points: a tangent method approach

• Published in: Journal of Physics A: Mathematical and Theoretical
• Main Authors: Di Francesco, Philippe, Guitter, Emmanuel
• 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, Illinois State University, Illinois State University, ANR-14-CE25-0014,GRAAL,GRaphes et Arbres ALéatoires(2014)
• Format: Article in Journal/Newspaper
• Language: English
• Published: HAL CCSD 2018
• Subjects: [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]; Arctic
• Online Access: https://cea.hal.science/cea-02011867; https://cea.hal.science/cea-02011867/document; https://cea.hal.science/cea-02011867/file/gUITT.pdf; https://doi.org/10.1088/1751-8121/aad028

International audience We use the tangent method of Colomo and Sportiello to investigate the arctic curve in a model of non-intersecting lattice paths with arbitrary fixed starting points aligned along some boundary and whose distribution is characterized by some arbitrary piecewise differentiable function. We find that the arctic curve has a simple explicit parametric representation depending of this function, providing us with a simple transform that maps the arbitrary boundary condition to the arctic curve location. We discuss generic starting point distributions as well as particular freezing ones which create additional frozen domains adjacent to the boundary, hence new portions for the arctic curve. A number of examples are presented, corresponding to both generic and freezing distributions. Our results corroborate already known expressions obtained by more involved methods based on bulk correlations, hence providing more evidence to the validity of the tangent method.