This should really be a comment, but I don't have any rep yet. First off, what kind of results do you want to know? Were you studying the compact case from Buser? I thought he included references/comments about analogues in the non-compact cases but I may be mistaken.
Anyway, you could try Sarnak's Bull. AMS article "Spectra of Hyperbolic Surfaces" which discusses some results. He also has a Notices article on Selberg's eigenvalue conjecture if you are interested in that. I can't think of anything else right now, and know nothing about the probabilistic results you seek.
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