# Short geodesic loops and $$L^p$$ L p norms of eigenfunctions on large genus random surfaces

@article{Gilmore2019ShortGL, title={Short geodesic loops and \$\$L^p\$\$ L p norms of eigenfunctions on large genus random surfaces}, author={Clifford Gilmore and Etienne Le Masson and Tuomas Sahlsten and Joe Thomas}, journal={Geometric and Functional Analysis}, year={2019}, volume={31}, pages={62-110} }

We give upper bounds for $$L^p$$ L p norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus $$g \rightarrow +\infty $$ g → + ∞ , we show that random hyperbolic surfaces X with respect to the Weil-Petersson volume have with high probability at most one such loop of length less than $$c \log g$$ c log g for small enough $$c > 0$$ c > 0 . This allows… Expand

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