William Hide

Short geodesics and small eigenvalues on random hyperbolic punctured spheres

We study the geometry and spectral theory of random genus 0 hyperbolic surfaces with n cusps in the regime of n tending to infinity. We look at counting the number of geodesics with lengths at scales roughly 1/sqrt(n). Sampling surfaces with respect to the Weil-Petersson probability measure we demonstrate Poisson statistics for the associated random variable in the large n limit. As a consequence, we gain an understanding of the systole of a typical genus 0 hyperbolic surface with many cusps.
Using similar methods we show that a genus 0 hyperbolic surface has o(n^1/3) small Laplacian eigenvalues with probability tending to 1 as n tends to infinity.
We shall compare and contrast these results to what is known for Weil-Petersson random compact surfaces in the large genus limit.
This is joint work with Joe Thomas (Durham).