I'm not sure if this is what you're looking for but there's a sizable literature on the 3-SAT phase transition.
Monasson, Zecchina, Kirkpatrcik, Selman and Troyansky had a paper in nature that talks about the phase transition of random k-SAT. They used a parameterization of the ratio of clauses to variables. For random 3-SAT, they found numerically that the transition point is around 4.3. Above this point random 3-SAT instances are over constrained and almost surely unsatsifiable and below this point problems are under constrained and satisfiable (with high probability). Mertens, Mezard and Zecchina use cavity method procedures to estimate the phase transition point to a higher degree of accuracy.
Far away from the critical point, "dumb" algorithms work well for satisfiable instances (walk sat, etc.). From what I understand, deterministic solver run times grow exponentially at or near the phase transition (see here for more of a discussion?).
A close cousin of belief propagation, Braunstein, Mezard and Zecchina have introduced survey propagation that is reported to solve satisfiable 3-SAT instances in millions of variables, even extremely close to the phase transition. Mezard has a lecture here on spin glasses (the theory of which he has used in analysis of random NP-Complete phase transitions) and Maneva has a lecture here on survey propagation.
From the other direction, it still looks like our best solvers take exponential amount of time to prove unsatisfiability. See here, here and here for proofs/discussion of the exponential nature of some common methods in proving unsatisfiability (Davis-Putnam procedures and resolution methods).
One has to be very careful about claims of 'easiness' or 'hardness' for random NP-Complete problems. Having an NP-Complete problem display a phase transition gives no guarantee as to where the hard problems are or whether there even are any. For example, the Hamiltoniain Cycle problem on Erdos-Renyi random graphs is provably easy even at or near the critical transition point. The Number Partition Problem doesn't seem to have any algorithms that solve it well into the probability 1 or 0 range, let alone near the critical threshold. From what I understand, random 3-SAT problems have algorithms that work well for satisfiable instances nearly at or below the critical threshold (survey propagation, walk sat, etc.) but no efficient algorithms above the critical threshold to prove unsatisfiability. This is just state of the art right now and could of course change in the future.