Singularly perturbed discrete differential equations

TL;DR

This paper studies singularly perturbed discrete differential equations with a parameter that changes the order, demonstrating a smooth transition and applying the findings to pattern counting in planar maps with probabilistic results.

Contribution

It introduces a framework for understanding smooth transitions in singularly perturbed discrete differential equations and applies it to combinatorial map pattern analysis.

Findings

Established conditions for smooth transition in singular perturbations

Derived a central limit theorem for non-self-intersecting pattern counts

Applied theoretical results to planar map enumeration

Abstract

Discrete differential equations appear most prominently in planar map and lattice path enumeration. In this work we consider discrete differential equations with an additional parameter $x$, where the order of the equation is $1$ for $x=0$ but $k> 1$ for $x\ne 0$. We call such equations singularly perturbed. The main contribution of this work is to show that there is actually a smooth transition under certain natural assumptions. As an application of this result we consider pattern counts in triangular planar maps and derive a central limit theorem for patterns which cannot self-intersect.