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

TL;DR

This paper derives an explicit formula for the arctic curve in a q-weighted non-intersecting lattice path model with arbitrary starting points, analyzing how the curve deforms with q and in various limits.

Contribution

It introduces a tangent method approach to obtain the arctic curve for models with arbitrary starting points and q-dependent weights, providing explicit formulas and detailed examples.

Findings

Explicit arctic curve expression in terms of starting point distribution and q

Analysis of arctic curve deformation as q varies

Limiting shapes of the arctic curve as q approaches 0 or infinity

Abstract

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.