Benjamini-Schramm limits of high genus translation surfaces: research announcement

TL;DR

This paper proves that high genus random translation surfaces, distributed according to Masur-Smillie-Veech measures, converge locally to a Poisson translation plane as genus increases, revealing new geometric limit behaviors.

Contribution

It introduces the concept of Benjamini-Schramm convergence for high genus translation surfaces and constructs a new Poisson translation plane as the local limit.

Findings

Convergence of random translation surfaces to a Poisson translation plane

Bounds on local geometric properties like injectivity radius probabilities

Introduction of a new random pointed surface model

Abstract

We prove that the sequence of Masur-Smillie-Veech (MSV) distributed random translation surfaces, with area equal to genus, Benjamini-Schramm converges as genus tends to infinity. This means that for any fixed radius $r>0$, if $X_g$ is an MSV-distributed random translation surface with area $g$ and genus $g$, and $o$ is a uniformly random point in $X_g$, then the radius-$r$ neighborhood of $o$ in $X_g$, as a pointed measured metric space, converges in distribution to the radius $r$ neighborhood of the root in a Poisson translation plane, which is a random pointed surface we introduce here. Along the way, we obtain bounds on statistical local geometric properties of translation surfaces, such as the probability that the random point $o$ has injectivity radius at most $r$, which may be of independent interest.