Spectral gap for Weil-Petersson random surfaces with cusps

2021/07/30 by Will Hide, Hide, Will · 1 citation
Mathematics · #05C50 #58J50 #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2107.14555

openalex publication_date 2021/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that for any ε>0, α∈[0,(1)/(2)), as g→∞ a generic finite-area genus g hyperbolic surface with n=O(g^α) cusps, sampled with probability arising from the Weil-Petersson metric on moduli space, has no non-zero eigenvalue of the Laplacian below (1)/(4)-((2α+1)/(4))²-ε. For α=0 this gives a spectral gap of size (3)/(16)-ε and for any α<(1)/(2) gives a uniform spectral gap of explicit size.

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