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 \(\Vert \varvec{p}_n\Vert _{\infty }=o_n\left( {t}^{-\frac{k}{2k-1}}\cdot \log ^{-\frac{k-1}{2k-1}}{t}\right) \).