Point Processes and Gap Statistics for Translation Surfaces

Abstract

We study the behavior of saddle connection directions on translation surfaces from the point process viewpoint. We define a direction point process for saddle connections of length at most $R>0$ on every translation surface. Given a general ergodic $\mathrm{SL}(2,\mathbb R)$-invariant measure $μ$ on a stratum of translation surfaces, we also define a slope point process associated with $μ$. By utilizing the dynamics and geometry of strata, we show that for almost every translation surface, the direction point processes, rescaled by $R^2$, converge in distribution to the slope point process as $R\to\infty$. Using Palm distribution theory, we prove the existence of limiting joint gap distributions and obtain new regularity results for gap distributions associated with the general $μ$. Finally, we study how these point processes and joint gap distributions behave as the genus $g$ tends to infinity. In the principal stratum, we show that the slope point processes, rescaled by $g^2$, converge in distribution to a Poisson point process. By passing to the corresponding Palm distributions, we show that the rescaled slope gaps converge jointly to independent exponential random variables, which is in stark contrast to the known non-exponential gap distributions in a fixed genus.