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

TL;DR

This paper employs the tangent method to derive explicit parametric formulas for arctic curves in non-intersecting lattice path models with arbitrary boundary conditions, including freezing scenarios, revealing how boundary distributions influence the arctic boundary.

Contribution

It introduces a simple transform linking arbitrary boundary conditions to arctic curves and extends analysis to freezing distributions creating additional frozen regions.

Findings

Explicit parametric representation of arctic curves for arbitrary boundary conditions

Transform linking boundary distribution to arctic curve location

Identification of frozen domains adjacent to boundaries in specific cases

Abstract

We use the tangent method 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.