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Asymptotic probability for connectedness

Thierry Monteil, Khaydar Nurligareev

Abstract

We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of other derivative combinatorial classes. The general result applies to rapidly growing combinatorial structures, which we call gargantuan, that also admit a sequence decomposition. The result is then applied to several models of graphs, of surfaces (square-tiled surfaces, combinatorial maps), and to geometric models of higher dimension (constellations, graph encoded manifolds). The corresponding derivative combinatorial classes are irreducible (multi)tournaments, indecomposable (multi)permutations and indecomposable perfect (multi)matchings.

Asymptotic probability for connectedness

Abstract

We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of other derivative combinatorial classes. The general result applies to rapidly growing combinatorial structures, which we call gargantuan, that also admit a sequence decomposition. The result is then applied to several models of graphs, of surfaces (square-tiled surfaces, combinatorial maps), and to geometric models of higher dimension (constellations, graph encoded manifolds). The corresponding derivative combinatorial classes are irreducible (multi)tournaments, indecomposable (multi)permutations and indecomposable perfect (multi)matchings.
Paper Structure (17 sections, 17 theorems, 96 equations)

This paper contains 17 sections, 17 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Tools
  3. Asymptotic expansion
  4. Combinatorial classes and decompositions
  5. Gargantuan classes and Bender theorem
  6. Double SET/SEQ decomposition
  7. Main results
  8. Asymptotics for p-periodic sequences
  9. Applications
  10. Graphs
  11. Surfaces
  12. Square-tiled surfaces
  13. $p$-angulations and combinatorial maps
  14. Higher dimensions
  15. Graph Encoded Manifolds
  16. ...and 2 more sections

Key Result

Theorem \ref{theorem:SET-asymptotics}

Let $\mathcal{A}$ be a gargantuan labeled combinatorial class with positive counting sequence, such that $\mathcal{A} = \mathrm{\sc SET}(\mathcal{C}) = \hbox{\sc SEQ}(\mathcal{D})$ for some labeled combinatorial classes $\mathcal{C}$ and $\mathcal{D}$. Suppose that $a\in\mathcal{A}$ is a random obje

Theorems & Definitions (47)

  • Theorem \ref{theorem:SET-asymptotics}: $\mathrm{\sc SET}$ asymptotics
  • Corollary \ref{proposition:graph}
  • Definition \ref{proposition:graph}
  • Theorem \ref{proposition:graph}: Bender Bender1975, in the simplified form of Odlyzko1995
  • Lemma \ref{proposition:graph}
  • proof
  • Lemma \ref{proposition:graph}
  • proof
  • Definition \ref{proposition:graph}
  • Theorem \ref{proposition:graph}: $\mathrm{\sc SET}$ asymptotics
  • ...and 37 more