The tight length spectrum of large-genus random hyperbolic surfaces with many cusps

Abstract: Since the work of Mirzakhani & Petri \cite{Mirzakhani_petri_2019} on random hyperbolic surfaces of large genus, length statistics of closed geodesics have been studied extensively. We focus on the case of random hyperbolic surfaces with cusps, the number $n_g$ of which grows with the genus $g$. We prove that if $n_g$ grows fast enough and we restrict attention to special geodesics that are \emph{tight}, we recover upon proper normalization the same Poisson point process in the large-$g$ limit for the length statistics. The proof relies on a recursion formula for tight Weil-Petersson volumes obtained in \cite{budd2023topological} and on a generalization of Mirzakhani's integration formula to the tight setting.