We give upper bounds for \(L^p\) 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 \(g \rightarrow +\infty \), 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 \(c \log g\) for small enough \(c > 0\). This allows us to deduce that the \(L^p\) norms of \(L^2\) normalised eigenfunctions on X are \(O(1/\sqrt{\log g})\) with high probability in the large genus limit for any \(p > 2 + \varepsilon \) for \(\varepsilon > 0\) depending on the spectral gap \(\lambda _1(X)\) of X, with an implied constant depending on the eigenvalue and the injectivity radius.

我们根据参数通过通过点的短测地线环数量的增长率来给出紧双曲表面上拉普拉斯算子的本征函数\（L ^ p \）范数的上限。当类\（g \ rightarrow + \ infty \）时，我们证明相对于Weil-Petersson体积的随机双曲曲面X最多有一个这样的长度小于\（c \ log g \）的循环对于足够小的\（c> 0 \）。这使我们可以推断出X上\（L ^ 2 \）归一化本征函数的\（L ^ p \）范数是\（O（1 / \ sqrt {\ log g}）\）与在大属极限高的概率为任何\（P> 2 + \ varepsilon \）为\（\ varepsilon> 0 \）取决于频谱空隙\（\拉姆达_1（X）\）的X，有一个隐含的取决于特征值和内射半径。