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

Philippe Di Francesco (1, 2, 3) , Emmanuel Guitter (2, 3)

1 Department of Mathematics, Illinois State University
2 IPHT - Institut de Physique Théorique - UMR CNRS 3681
3 Université Paris-Saclay

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.

Mots clés: non-intersecting lattice paths; continuum limit; arctic curve

Domaines: Physique [physics]

Philippe Di Francesco, Emmanuel Guitter. A tangent method derivation of the arctic curve for $q$-weighted paths with arbitrary starting points. Journal of Physics A: Mathematical and Theoretical, 2019, 52 (11), pp.115205. ⟨10.1088/1751-8121/ab03ff⟩. ⟨cea-02932285⟩