A tree bijection for the moduli space of genus-0 hyperbolic surfaces with boundaries

The Weil-Petersson volume of genus-g hyperbolic surfaces with geodesic boundaries is known since work of Mirzakhani to be polynomial in the boundary lengths. We provide a bijective proof of this fact in the genus-0 case in the presence of a distinguished cusp. It is based on a generalization of a recent tree bijection, by the first author and Curien, to the setting with geodesic boundaries, requiring an extension of the Bowditch-Epstein-Penner spine construction. As an application of our tree bijection we establish an explicit formula for the distance-dependent three-point function, which records an exact metric statistic measuring the difference of two geodesic distances among a triple of distinguished cusps in a Weil-Petersson random surface. We conclude with a discussion of the relevance of this function to the topological recursion of Weil-Petersson volumes and metric properties of Weil-Petersson random surfaces with many boundaries or cusps.