I will talk about the model of "random hyperbolic surfaces", picked according to the Weil-Petersson measure. I will be interested in the spectrum of the laplacian on these random objects, in the large volume asymptotic. I will in particular discuss the spectral gap, and will report on joint work with Laura Monk, aiming to show that the lowest eigenvalue of a random compact hyperbolic surfaces is typically greater than 1/4-ε.

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