arXiv subjects

Bart Zonneveld

Publications and source records attributed to Bart Zonneveld.

4 recordsLinked to original sources

arXiv2025

A tree bijection for cusp-less planar hyperbolic surfaces

Recently, a tree bijection has been found for planar hyperbolic surfaces, which allows for an easy computation of the Weil--Petersson volumes, and opens the path to get distance statistic on random hyperbolic surfaces and to find scaling limits when the number of boundaries becomes large. Crucially, this tree bijection requires the hyperbolic surface to have at least one cusp as origin, from which point distances are measured. In this paper we will extend this tree bijection, such that having a cusp is no longer required. We will first extend the bijection to half-tight cylinders. Since general planar hyperbolic surfaces can be naturally decomposed in two half-tight cylinders, this general case is also covered. In the half-tight cylinder the distances to the origin are replaced by the so-called Busemann function. This Busemann function is not well-defined on the surface, but it is on the cylinder cover.

math.GT↗

arXiv2025

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.

math.GT↗

arXiv2025

A tree bijection for rigid quadrangulations

We study the counting problem of rigid quadrangulations, recently introduced by Budd and proven to be in bijection with colorful quadrangulations. The generating function for the latter has been derived in an algebraic manner by Bousquet-Mélou and Elvey Price, which therefore also counts rigid quadrangulations. In this paper we will provide a direct, bijective proof, for this generating function. We will relate the rigid quadrangulations to some naturally appearing trees, decorated with certain natural data. By some slight bijective manipulation of the data, we get a decorated tree, for which the generating function can be found. This result opens the door to better understand the geometry of random rigid quadrangulations (and maybe even of random colorful quadrangulations), by studying the corresponding decorated trees. These properties are relevant the formulation of UV complete JT-gravity, following the work of Ferrari.

math.CO↗

arXiv2023

Topological recursion of the Weil-Petersson volumes of hyperbolic surfaces with tight boundaries

The Weil-Petersson volumes of moduli spaces of hyperbolic surfaces with geodesic boundaries are known to be given by polynomials in the boundary lengths. These polynomials satisfy Mirzakhani's recursion formula, which fits into the general framework of topological recursion. We generalize the recursion to hyperbolic surfaces with any number of special geodesic boundaries that are required to be tight. A special boundary is tight if it has minimal length among all curves that separate it from the other special boundaries. The Weil-Petersson volume of this restricted family of hyperbolic surfaces is shown again to be polynomial in the boundary lengths. This remains true when we allow conical defects in the surface with cone angles in $(0,π)$ in addition to geodesic boundaries. Moreover, the generating function of Weil-Petersson volumes with fixed genus and a fixed number of special boundaries is polynomial as well, and satisfies a topological recursion that generalizes Mirzakhani's formula. We will comment on a possible geometric interpretation of this formula. This work is largely inspired by recent works by Bouttier, Guitter & Miermont on the enumeration of planar maps with tight boundaries. Our proof relies on the equivalence of Mirzakhani's recursion formula to a sequence of partial differential equations (known as the Virasoro constraints) on the generating function of intersection numbers. Finally, we discuss a connection with JT gravity. We show that the multi-boundary correlators of JT gravity with defects (cone points or FZZT branes) are expressible in the tight Weil-Petersson volume generating functions, using a tight generalization of the JT trumpet partition function.

math-ph↗