I will define random surfaces X associated to arrangements of curves A. The main relation between X and A is that they asymptotically share proportional Chern invariants. I use a large scale behavior of Dedekind sums and lengths of continued fractions to prove that relation. It turns out that a random ingredient is necessary for the asymptotic result. The main application of random surfaces is to explore the geography of simply connected surfaces. It is an old open question to know if there are additional contrains on Chern invariants of simply connected smooth projective surfaces of general type. I will explain the problem and the new numbers provided by random surfaces.

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