# Random sparse bit strings at the threshold of adjacency

Complexity I

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## 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 (where

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

*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*σ*Ø 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*.### Keywords

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© Springer-Verlag 1998