Typical hyperbolic surfaces under the Weil-Petersson measure have spectral gap at least 2/9 - ε.
Towards large genus asymptotics of intersection numbers on moduli spaces of curves.Geometric and Functional Analysis, 25(4):1258–1289
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Typical hyperbolic surfaces have a spectral gap greater than $2/9 - \epsilon$
Typical hyperbolic surfaces under the Weil-Petersson measure have spectral gap at least 2/9 - ε.