Limiting probabilities of first order properties of random sparse graphs and hypergraphs

TL;DR

This paper investigates the limiting probabilities of first order properties in sparse random graphs and hypergraphs, revealing a phase transition at a critical average degree where the set of possible limiting probabilities changes from missing intervals to covering the entire range.

Contribution

It establishes a phase transition at a critical value for the closure of limiting probabilities of first order properties in sparse graphs and hypergraphs.

Findings

For $c \,\geq\, c_0 \approx 0.93$, the closure of limiting probabilities covers [0,1].

For $c < c_0$, the closure misses at least one subinterval.

Results extend to random hypergraphs with hyperedge probability $p=c/n^{d-1}$.

Abstract

Let $G_n$ be the binomial random graph $G(n,p=c/n)$ in the sparse regime, which as is well-known undergoes a phase transition at $c=1$. Lynch (Random Structures Algorithms, 1992) showed that for every first order sentence $\phi$, the limiting probability that $G_n$ satisfies $\phi$ as $n\to\infty$ exists, and moreover it is an analytic function of $c$. In this paper we consider the closure $\overline{L_c}$ in $[0,1]$ of the set $L_c$ of all limiting probabilities of first order sentences in $G_n$. We show that there exists a critical value $c_0 \approx0.93$ such that $\overline{L_c}= [0,1]$ when $c \ge c_0$, whereas $\overline{L_c}$ misses at least one subinterval when $c<c_0$. We extend these results to random $d$-uniform sparse hypergraphs, where the probability of a hyperedge is given by $p=c/n^{d-1}$.