Biased random k-SAT

The basic random $k$-SAT problem is: Given a set of $n$ Boolean variables, and $m$ clauses of size $k$ picked uniformly at random from the set of all such clauses on our variables, is the conjunction of these clauses satisfiable? Here we consider a variation of this problem where there is a bias towards variables occurring positive -- i.e. variables occur negated w.p. $0<p< \frac{1}{2}$ and positive otherwise -- and study how the satisfiability threshold depends on $p$. For $p<\frac{1}{2}$ this model breaks many of the symmetries of the original random $k$-SAT problem, e.g. the distribution of satisfying assignments in the Boolean cube is no longer uniform. For any fixed $k$, we find the asymptotics of the threshold as $p$ approaches $0$ or $\frac{1}{2}$. The former confirms earlier predictions based on numerical studies and heuristic methods from statistical physics.