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Thanks for reading my question ! Could anybody give me some references ( books, papers containing elementary results etc ) on the eigen values and eigenspectra of NON-compact Riemann surfaces. I studied the compact cases briefly and want to know the analogues or further results for the non-compact ones.

Also, I need some references on probabilistic results on compact or non-compact Riemann surfaces, like random walks and Brownian motion on them etc.

Thank you !

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up vote 2 down vote accepted

If your French is alright, Nicolas Bergeron has written a beautiful book, Le spectre des surfaces hyperboliques, available here, which covers many of these topics in detail.

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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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