On the concentration of the number of solutions of random satisfiability formulas

Let $Z(F)$ be the number of solutions of a random $k$-satisfiability formula $F$ with $n$ variables and clause density $\alpha$. Assume that the probability that $F$ is unsatisfiable is $O(1\mathrm{/}\log (n{)}^{1+\text{e}})$ for $\text{e}>0$. We show that (possibly excluding a countable set of `exceptional’ $\alpha$’s) the number of solutions concentrate in the logarithmic scale, i.e., there exists a non-random function $\varphi (\alpha )$ such that, for any $\delta >0$, $(1/n)\log Z(F)\in [\phi-\delta,\phi+\delta]$ with high probability. In particular, the assumption holds for all $\alpha <1$, which proves the above concentration claim in the whole satisfiability regime of random $2$-SAT. We also extend these results to a broad class of constraint satisfaction problems. The proof is based on an interpolation technique from spin-glass theory, and on an application of Friedgut’s theorem on sharp thresholds for graph properties.