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

Let (Formula presented.) be the binomial random graph (Formula presented.) in the sparse regime, which as is well-known undergoes a phase transition at (Formula presented.). Lynch ((Formula presented.), 1992) showed that for every first order sentence (Formula presented.), the limiting probability that (Formula presented.) satisfies (Formula presented.) as (Formula presented.) exists, and moreover it is an analytic function of c. In this paper we consider the closure (Formula presented.) in the interval (Formula presented.) of the set (Formula presented.) of all limiting probabilities of first order sentences in (Formula presented.). We show that there exists a critical value (Formula presented.) such that (Formula presented.) when (Formula presented.), whereas (Formula presented.) misses at least one subinterval when (Formula presented.). We extend these results to random sparse d-uniform hypergraphs, where the probability of a d-edge is (Formula presented.)