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

We give upper bounds for $L^p$ norms of eigenfunctions the Laplacian on compact hyperbolic surfaces in terms a parameter depending growth rate number short geodesic loops passing through point. When genus $g \to +\infty$, we show that random $X$ with respect to Weil-Petersson volume have high probability at most one such loop length less than $c \log g$ small enough > 0$. This allows us deduce $L^2$ normalised are $O(1/\sqrt{\log g})$ large limit any $p 2 + \varepsilon$ $\varepsilon 0$ spectral gap $\lambda_1(X)$ $X$, an implied constant eigenvalue and injectivity radius.