Sharpness of the Satisfiability Threshold for Non-uniform Random k-SAT

Abstract

We study non-uniform random k-SAT on n variables with an arbitrary probability distribution $\varvec{p}$ on the variable occurrences. The number $t=t(n)$ of randomly drawn clauses at which random formulas go from asymptotically almost surely (a. a. s.) satisfiable to a. a. s. unsatisfiable is called the satisfiability threshold. Such a threshold is called sharp if it approaches a step function as n increases. We show that a threshold t(n) for random k-SAT with an ensemble $(\varvec{p}_n)_{n\in \mathbb {N}}$ of arbitrary probability distributions on the variable occurrences is sharp if $\Vert \varvec{p}_n\Vert _2^2=\mathcal {O}_n\left( {t}^{-\frac{2}{k}}\right)$ and $\Vert \varvec{p}_n\Vert _{\infty }=o_n\left( {t}^{-\frac{k}{2k-1}}\cdot \log ^{-\frac{k-1}{2k-1}}{t}\right)$.

This result generalizes Friedgut’s sharpness result from uniform to non-uniform random k-SAT and implies sharpness for thresholds of a wide range of random k-SAT models with heterogeneous probability distributions, for example such models where the variable probabilities follow a power-law distribution.