  An analysis of the average properties of a local search resolution procedure
for the satisfaction of random Boolean constraints is presented. Depending on
the ratio alpha of constraints per variable, resolution takes a time T_res
growing linearly (T_res \sim tau(alpha) N, alpha < alpha_d) or exponentially
(T_res \sim exp(N zeta(alpha)), alpha > alpha_d) with the size N of the
instance. The relaxation time tau(alpha) in the linear phase is calculated
through a systematic expansion scheme based on a quantum formulation of the
evolution operator. For alpha > alpha_d, the system is trapped in some
metastable state, and resolution occurs from escape from this state through
crossing of a large barrier. An annealed calculation of the height zeta(alpha)
of this barrier is proposed. The polynomial/exponentiel cross-over alpha_d is
not related to the onset of clustering among solutions.
