Asymptotic probability for connectedness

TL;DR

This paper investigates the asymptotic probability of connectedness in combinatorial objects, revealing integer coefficients linked to derivative classes, with applications across graph, surface, and geometric models.

Contribution

It introduces a general framework for the asymptotic expansion of connectedness probabilities and interprets coefficients as counting sequences of derivative combinatorial classes.

Findings

Coefficients in asymptotics are integers.

Applicable to various combinatorial structures including graphs and surfaces.

Derived classes are irreducible or indecomposable, aiding enumeration.

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.