Uniform asymptotics of area-weighted Dyck paths

TL;DR

This paper derives a uniform asymptotic expression for the bivariate generating function of Dyck paths weighted by length and area, using advanced steepest descent methods near a tricritical point.

Contribution

It introduces a novel application of the generalized steepest descent method to obtain uniform asymptotics for Dyck paths with area and length weights.

Findings

Asymptotic expression valid near the tricritical point.

Uniform approximation across a range of length variables.

Extension of steepest descent techniques to coalescing saddle points.

Abstract

Using the generalized method of steepest descents for the case of two coalescing saddle points, we derive an asymptotic expression for the bivariate generating function of Dyck paths, weighted according to their length and their area in the limit of the area generating variable tending towards 1. The result is valid uniformly for a range of the length generating variable, including the tricritical point of the model.