Phase transitions in graphs on orientable surfaces

TL;DR

This paper investigates the component structure of random graphs embeddable on orientable surfaces, revealing two distinct phase transitions in their emergence and growth, which differ from classical Erdős–Rényi graphs.

Contribution

It establishes the existence of two phase transitions in graphs on orientable surfaces and provides asymptotic estimates for their counts across these regimes.

Findings

First phase transition at m=n/2+O(n^{2/3}) with emergence of a giant component.

Second phase transition at m=n+O(n^{3/5}) where the giant covers almost all vertices.

Distinct behavior from Erdős–Rényi graphs observed on surfaces.

Abstract

Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We prove that the component structure of a graph chosen uniformly at random from the class $\mathcal{S}_g(n,m)$ of all graphs on vertex set $[n]=\{1,\dotsc,n\}$ with $m$ edges embeddable on $\mathbb{S}_g$ features two phase transitions. The first phase transition mirrors the classical phase transition in the Erd\H{o}s--R\'enyi random graph $G(n,m)$ chosen uniformly at random from all graphs with vertex set $[n]$ and $m$ edges. It takes place at $m=\frac{n}{2}+O(n^{2/3})$, when a unique largest component, the so-called \emph{giant component}, emerges. The second phase transition occurs at $m = n+O(n^{3/5})$, when the giant component covers almost all vertices of the graph. This kind of phenomenon is strikingly different from $G(n,m)$ and has only been observed for graphs on surfaces. Moreover, we derive an asymptotic estimation of the number of graphs in $\mathcal{S}_g(n,m)$ throughout the regimes of these two phase transitions.