The systole of random hyperbolic 3-manifolds

TL;DR

This paper investigates the behavior of the shortest non-contractible loop (systole) in large random hyperbolic 3-manifolds, providing a limit formula and numerical estimates for its expected value.

Contribution

It establishes the existence of a limit for the expected systole in random hyperbolic 3-manifolds and derives a closed-form expression for it.

Findings

The expected systole converges as volume increases.

A closed formula for the limit of the expected systole is provided.

Numerical approximations of the limit are computed.

Abstract

We study the systole of a model of random hyperbolic 3-manifolds introduced by Petri and Raimbault, answering a question posed in that same article. These are compact manifolds with boundary constructed by randomly gluing truncated tetrahedra along their faces. We prove that the limit, as the volume tends to infinity, of the expected value of their systole exists and we give a closed formula of it. Moreover, we compute a numerical approximation of this value.