Random large genus closed hyperbolic surfaces have first Laplacian eigenvalue approaching the optimal 1/4 with high probability, as proved by Anantharaman and Monk and surveyed in this paper.
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Chaos in large genus surfaces
Random large genus closed hyperbolic surfaces have first Laplacian eigenvalue approaching the optimal 1/4 with high probability, as proved by Anantharaman and Monk and surveyed in this paper.