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Asymptotic normality of pattern counts in random maps II

Eva-Maria Hainzl

Abstract

In a recent work, a central limit theorem for pattern counts in random planar maps was proven by reducing the problem to a face count problem. We provide a shorter proof by circumventing this reduction through the computation of bivariate coefficient asymptotics from a functional equation with one catalytic variable and extend the result to pattern counts with arbitrary boundary and new map classes.

Asymptotic normality of pattern counts in random maps II

Abstract

In a recent work, a central limit theorem for pattern counts in random planar maps was proven by reducing the problem to a face count problem. We provide a shorter proof by circumventing this reduction through the computation of bivariate coefficient asymptotics from a functional equation with one catalytic variable and extend the result to pattern counts with arbitrary boundary and new map classes.
Paper Structure (12 sections, 15 theorems, 59 equations, 3 figures)

This paper contains 12 sections, 15 theorems, 59 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A simplified proof of asymptotic normality of pattern counts
  3. Extension to further map classes
  4. Bipartite maps
  5. 2-connected maps
  6. Proofs
  7. Proof of Lemma \ref{['lem:S1']}
  8. Proof of Lemma \ref{['lem:mov_bi']}
  9. Proof of Lemma \ref{['lem:bivar']}
  10. Proof of Lemma \ref{['lem:moments']}
  11. Proof of Lemma \ref{['lem:partial']}
  12. Conclusion

Key Result

Theorem 1

Let $\textbf{p}$ be a planar map and $X_n$ the number of pattern occurrences of $\textbf{p}$ in a random planar map of size $n$. Then, where $\mu_n = c_1n$ and $\sigma_n = c_2\sqrt{n}$ for some computable constants $c_1,c_2 \geq 0$ as $n\rightarrow \infty$.

Figures (3)

  • Figure 1: Decomposition of planar maps M with labeled patterns intersecting at most pairwise
  • Figure 2: Left: a fly. The green arrows correspond to the rotations of the pattern occurrence. Right: Decomposition of map where the root edge is incident to a distinguished fly into the root edge and a map with simple partial boundary.
  • Figure 4: Decomposition of 2-connected maps

Theorems & Definitions (32)

  • Definition 1: Rooted planar maps
  • Definition 2: Pattern occurrences BenderGaoRichmond
  • Theorem 1
  • Theorem 2: Gao and Wormald, GaoWormald
  • Lemma 1
  • Definition 3: Intersection types, rotations and deep faces
  • Definition 4: (Partial) simple boundaries
  • Proposition 1
  • proof
  • Theorem 3: Drmota, Noy, Yu DrmotaNoyYu
  • ...and 22 more