Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces

TL;DR

This paper investigates the statistical fluctuations of self-intersection counts for random geodesics on negatively curved surfaces, revealing different fluctuation scales and distributions depending on whether the curvature is constant or variable.

Contribution

It establishes the order of fluctuations and their limiting distributions for self-intersection counts on negatively curved surfaces, distinguishing between constant and variable curvature cases.

Findings

Fluctuations are of order L for constant curvature, converging to a nondegenerate distribution.

Fluctuations are of order L^{3/2} for variable curvature, converging to a Gaussian.

Results apply to both closed geodesics and generic geodesics with random initial conditions.

Abstract

Let $\Upsilon $ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $\Upsilon$ of length $\leq L$, choose one, say $\gamma_{L}$, at random and let $N (\gamma_{L})$ be the number of its self-intersections. It is known that there is a positive constant $\kappa$ depending on the metric such that $N (\gamma_{L})/L^{2} \rightarrow \kappa$ in probability as $L\rightarrow \infty$. The main results of this paper concern the size of typical fluctuations of $N (\gamma_{L})$ about $\kappa L^{2}$. It is proved that if the metric has constant curvature -1 then typical fluctuations are of order $L$, in particular, $(N (\gamma_{L})-\kappa L^{2})/L$ converges weakly to a nondegenerate probability distribution. In contrast, it is also proved that if the metric has variable negative curvature then fluctuations of $N (\gamma_{L})$ are of order $L^{3/2}$, in particular, $(N (\gamma_{L})-\kappa L^{2})/L^{3/2}$ converges weakly to a Gaussian distribution. Similar results are proved for generic geodesics, that is, geodesics whose initial tangent vectors are chosen randomly according to normalized Liouville measure.