In zero knowledge proofs, would it make a difference if the prover is deterministic or probabilistic? In zero knowledge proofs, would it make a difference if the prover is deterministic or probabilistic?

As the verifier can simulate the prover's interaction with him, I assumed that the prover would need to be deterministic. Maybe this is a misunderstanding on my part?
In the general case of for auxiliary-input zero knowledge, it is impossible for the prover to be deterministic; see Definitions and properties of Zero-Knowledge proof systems by Goldreich and Oren. In the random oracle model, everything is different, so I don't want to go there. However, I will say that it is not true that the prover is deterministic is Fiat-Shamir. Fiat-Shamir is a way of converting some honest-verifier ZK protocols to fully malicious ones. Almost all honest-verifier ZK protocols considered have a probabilistic prover, so the Fiat-Shamir version also has a probabilistic prover.