  Let F be a uniformly distributed random k-SAT formula with n variables and m
clauses. Non-constructive arguments show that F is satisfiable for
clause/variable ratios m/n< r(k)~2^k ln 2 with high probability. Yet no
efficient algorithm is know to find a satisfying assignment for densities as
low as m/n r(k).ln(k)/k with a non-vanishing probability. In fact, the density
m/n r(k).ln(k)/k seems to form a barrier for a broad class of local search
algorithms. One of the very few algorithms that plausibly seemed capable of
breaking this barrier is a message passing algorithm called Belief Propagation
Guided Decimation. It was put forward on the basis of deep but non-rigorous
statistical mechanics considerations. Experiments conducted for k=3,4,5
suggested that the algorithm might succeed for densities very close to r_k.
Furnishing the first rigorous analysis of BP decimation, the present paper
shows that the algorithm fails to find a satisfying assignment already for
m/n>c.r(k)/k, for a constant c>0 (independent of k).
