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We study the number of short geodesics and small eigenvalues onWeil-Petersson random genus zero hyperbolic surfaces with $n$ cusps in theregime $n\to\infty$. Inspired by work of Mirzakhani and Petri \cite{Mi.Pe19},we show that the random multi-set of lengths of closed geodesics converges,after a suitable rescaling, to a Poisson point process with explicit intensity.As a consequence, we show that the Weil-Petersson probability that a hyperbolicpunctured sphere with $n$ cusps has at least $k=o(n)$ arbitrarily smalleigenvalues tends to $1$ as $n\to\infty$.

Short geodesics and small eigenvalues on random hyperbolic punctured spheres