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

Let Gn 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 (RandomStructuresandAlgorithms, 1992) showed that for every first order sentence phi, the limiting probability that Gn satisfies phi as n ->infinity exists, and moreover it is an analytic function of c. In this paper we consider the closure Lc? in the interval [0,1] of the set Lc of all limiting probabilities of first order sentences in Gn. We show that there exists a critical value c0 approximate to 0.93 such that Lc?=[0,1] when c >= c0, whereas Lc? misses at least one subinterval when c<c0. We extend these results to random sparse d-uniform hypergraphs, where the probability of a d-edge is p=c/nd-1.