The simple separating systole for hyperbolic surfaces of large genus

TL;DR

This paper demonstrates that for large genus hyperbolic surfaces, the expected separating systole grows logarithmically with genus, contrasting with the constant behavior of the systole.

Contribution

It establishes the asymptotic behavior of the expected separating systole for random hyperbolic surfaces of large genus.

Findings

Expected separating systole behaves like 2 log g as genus g increases.

Expected systole remains independent of genus, contrasting with separating systole.

Highlights difference in geometric properties of random surfaces at large genus.

Abstract

In this note we show that the expected value of the separating systole of a random surface of genus $g$ with respect to Weil-Petersson volume behaves like $2\log g $ as the genus goes to infinity. This is in strong contrast to the behavior of the expected value of the systole which, by results of Mirzakhani and Petri, is independent of genus.