What specific evidence is there for P = RP? RP is the class of problems decidable by a nondeterministic Turing machine that terminates in polynomial time, but that is also allowed one-sided error.  P is the usual class of problems decidable by a deterministic Turing machine that terminates in polynomial time.

P = RP follows from a relationship in circuit complexity.  Impagliazzo and Wigderson showed that P = BPP follows if some problem that can be decided in deterministic exponential time also requires exponential size circuits (note that P = BPP implies P = RP).  Perhaps due to these results, there seems to be a feeling among some complexity theorists that probabilistic reductions can probably be derandomized.


  What other specific evidence is there that P = RP?
The existence of problems in DTIME(2^O(n)) which require exponential-size circuits to compute (which is the assumption in IW) seems plausible since otherwise we would have non-uniformity giving a speedup on EVERY computational problem -- which goes completely against the current thinking that does not see a "too significant" gap between uniform and non-uniform complexity for "normal" problems.  This thinking comes from the fact that there are very few examples where a "non-uniform" algorithm is known that is significantly better than the known uniform one (again except for derandomization).

Another piece of "evidence" is that relative to a random oracle we do have P=BPP.