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Singularly perturbed discrete differential equations

Michael Drmota, Eva-Maria Hainzl

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.

Singularly perturbed discrete differential equations

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 , where the order of the equation is for but for . 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.
Paper Structure (5 sections, 3 theorems, 49 equations, 5 figures)

This paper contains 5 sections, 3 theorems, 49 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Singularly perturbed discrete differential equations
  3. Pattern counts in simple triangulations
  4. Conclusions
  5. References

Key Result

Theorem 1

Let $F(z,u,x) = \sum_{i\geq 0} f_i(z,x)u^i$ be implicitely given by the DDE (eqpertDDE), where $k\ge 2$, $Q(z,u,y_0,y_1)$ is rational with non-negative coefficients, $R(z,u,y_0,y_1,\ldots, y_k)$ rational but linear in $y_k$ and both with large enough region of convergence. Further, let $Q(z,0,0,0) \ then $X_n$ ($n \equiv j \bmod d$, $j\in J$) satisfies a central limit theorem of the form where $\

Figures (5)

  • Figure 1: The solutions $u_1(x), u_2(x)$ for the values $x= 0, 0.025, 0.05, 0.1$ in the first line and $x= -0.01, -0.025, -0.05, -0.1$ in the second line. The red dot marks the origin, the green line the negative value of $x$ to show that $0>u_2(z) > -x$ for $z>0$.
  • Figure 2: The solution $T(z,0,x)$ for the values $x= 0, 0.025, 0.05, 0.1$ and in the second line $x=-0.005,-0.01,-0.025, -0.05$. The red dot marks the origin.
  • Figure 3: Decomposition of triangulations with a root edge incident to a labeled pattern occurrence into a sequence of edges and multifan-triangulations
  • Figure 4: Decomposition of multifan-triangulations into an alternating sequence of near-triangulations incident to the pattern occurrence and sequences of near-triangulations
  • Figure 5: The diamond pattern which cannot self-intersect

Theorems & Definitions (8)

  • Theorem 1
  • Example 1
  • Definition 1
  • Definition 2: Pattern occurrences BenderGaoRichmond
  • Proposition 1
  • proof
  • Example 2: Diamonds
  • Theorem 2