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arXiv · 2603.19484

Singularly perturbed discrete differential equations

Michael Drmota·Eva-Maria Hainzl
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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.

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Michael Drmota, Eva-Maria Hainzl. 2026-03-19. Singularly perturbed discrete differential equations. https://arxiv.org/abs/2603.19484

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