What do we know about the phase transition of #P-Complete problems? What is known about the phase transition in #P-Complete problems? Specifically, does there exists a different phase transition for #DNF-k-SAT and #CNF-k-SAT? Update:
As we know, there is a phase transition in Random k-SAT where solving the problem goes from being easy to hard and back to easy again. I would like to know if there is such a phenomenon for #P-Complete problems as well. More importantly, if there exists a phase transition, is it same for #CNF-k-SAT and #DNF-k-SAT? I am thinking that there is some type of phase transition for #CNF-k-SAT. On the other hand, I do not think there is phase transition for #DNF-k-SAT and the problem gets harder as we add more clauses....
For counting independent sets, there is a recent proof for a computational phase transition, by Allan Sly:
http://arxiv.org/abs/1005.5584
(the algorithm is by Dror Weitz from 2006; Allan proved the matching hardness and co-won the best paper award in FOCS'10 for that)

Note that for random 3SAT and similar problems there is no proof that those problems are indeed hard in the appropriate interval. When you go to the harder counting problems, it becomes easier to prove the hardness.