The number of random 2-SAT solutions is asymptotically log-normal

TL;DR

This paper proves that in the satisfiable phase of random 2-SAT, the logarithm of the number of solutions follows a normal distribution with fluctuations of order sqrt(n), contrasting with other CSPs.

Contribution

It establishes a central limit theorem for the log of the number of solutions in random 2-SAT, providing explicit variance formulas and highlighting differences from other CSPs.

Findings

Logarithm of solutions satisfies a central limit theorem

Fluctuations are of order sqrt(n)

Variance can be explicitly computed

Abstract

We prove that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem. This implies that the log of the number of satisfying assignments exhibits fluctuations of order $\sqrt n$, with $n$ the number of variables. The formula for the variance can be evaluated effectively. By contrast, for numerous other random constraint satisfaction problems the typical fluctuations of the logarithm of the number of solutions are {\em bounded} throughout all or most of the satisfiable regime.