What is the satisfiability threshold of random balanced Boolean expressions?

We consider the model of random Boolean expressions based on balanced binary trees with 2N$$ {2}^N $$ leaves, to which are randomly attributed one of kN$$ {k}_N $$ Boolean variables or their negations. We prove that if for every c>0$$ c>0 $$ it holds that kNexp(−cN)→0$$ {k}_N\exp \left(-c\sqrt{N}\right)\to 0 $$ then asymptotically with high probability the Boolean expression is either a tautology or an antitautology. Our methods are based on the study of a certain binary operation on the set of probability measures on {0,1}I$$ {\left\{0,1\right\}}^I $$ for a finite set I.