Random sparse bit strings at the threshold of adjacency

Abstract

We give a complete characterization for the limit probabilities of first order sentences over sparse random bit strings at the threshold of adjacency. For strings of length n, we let the probability that a bit is “on” be $\tfrac{c}{{\sqrt n }}$, for a real positive number c. For every first order sentence Ø, we show that the limit probability function:

$$f_\phi (c) = \mathop {\lim }\limits_{n \to \infty } Pr[U_{n,\tfrac{c}{{\sqrt n }}} has the property \phi ]$$

(where U _n, $\tfrac{c}{{\sqrt n }}$ is the random bit string of length n) is infinitely differentiable. Our methodology for showing this is in itself interesting. We begin with finite models, go to the infinite (via the almost sure theories) and then characterize f _σ(c) as an infinite sum of products of polynomials and exponentials. We further show that if a sentence σØ has limiting probability 1 for some c, then σ has limiting probability identically 1 for every c. This gives the surprising result that the almost sure theories are identical for every c.