Friedrich, Tobias; Rothenberger, Ralf **Sharpness of the Satisfiability Threshold for Non-Uniform Random k-SAT**Theory and Applications of Satisfiability Testing (SAT) 2018: 273–291

We study non-uniform random k-SAT on n variables with an arbitrary probability distribution 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 \((p_n)_{n\in\mathbb{N}}\) of arbitrary probability distributions on the variable occurrences is sharp if \(\|p\|_2^2 = O_n(t^{-2/k})\) and \(\|p\|_\infty\) \(= o_n(t^{-k/(2k-1)} \log^{-(k-1)/(2k-1)}(t))\). 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.