Abstract:Let $X$ be a compact connected orientable hyperbolic surface and let $X_n$ be a degree $n$ cover, taken uniformly at random. We show that, with high probability, the $L^\infty$ norm of every Laplace eigenfunction on $X_n$ with bounded eigenvalue decays polynomially in $n$. This gives a polynomial decay analogue of the logarithmic bound of Gilmore--Le Masson--Sahlsten--Thomas [arXiv:1912.09961] in the Weil--Petersson model. Using similar methods, we also show that, with high probability, the distribution of eigenvalues of the Laplacian on $X_n$ converges to the spectral measure of the hyperbolic plane with polynomially decaying error. Our proof relies on the Selberg pre-trace formula and a variant of the polynomial method.
From: Elena Kim [view email]
[v1]
Sun, 1 Mar 2026 14:20:17 UTC (20 KB)
[v2]
Thu, 26 Mar 2026 10:32:38 UTC (27 KB)
[v3]
Fri, 28 Aug 2026 13:55:48 UTC (40 KB)
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