Assuming the cryptographic hash function meets its design goals, the output will (provably) follow a uniform distribution over its period, as every input to the hash function is unique by design.
One of the goals of a hash function is to approximate a random oracle, that is, for any two distinct inputs A and B, the outputs H(A) and H(B) should (for a true random oracle) be uncorrelated. Hash functions get pretty close to that, but of course weaknesses creep in with time and cryptanalysis.
That said, cryptographic primitives are essentially the best mathematical algorithms we have available when it comes to quality, therefore it is safe to say that if they cannot solve your problem, nothing will.