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Counting statistics for geodesics on flat surfaces

Stephen Cantrell, Mark Pollicott

TL;DR

The paper develops Dolgopyat-free counting limit laws for comparing geometric length and combinatorial costs on infinite graphs and applies these results to translation surfaces, where saddle-connection paths yield two key quantities: the geometric length $\ell(p)$ and the singularity-visit length $|p|$. Through a complex-analytic and spectral framework, it establishes a law of large numbers, large deviations, and multi-dimensional central limit theorems for costs on path sets, under natural hypotheses (G1)–(G3) and their translation analogues (T1)–(T3). A central contribution is a method to center fluctuations via reparameterized transfer operators, derive asymptotics from a two-variable complex function $\eta_{\mathcal{G}}(s,t)$, and obtain non-degenerate Gaussian limits with explicit covariance structures. The results extend to holonomy vectors and visit counts to singularities, and a positive variance criterion ensures the CLTs are non-degenerate; this provides a rigorous probabilistic understanding of how geometric length and saddle-connection combinatorics relate on flat surfaces, with potential implications for quantitative geometry and Teichmüller dynamics.

Abstract

We study counting limit laws that compare length functions on infinite graphs. We then apply these results to flat surfaces to obtain a statistical comparison between the geometric length and the number of singularities visited by geodesic paths.

Counting statistics for geodesics on flat surfaces

TL;DR

The paper develops Dolgopyat-free counting limit laws for comparing geometric length and combinatorial costs on infinite graphs and applies these results to translation surfaces, where saddle-connection paths yield two key quantities: the geometric length and the singularity-visit length . Through a complex-analytic and spectral framework, it establishes a law of large numbers, large deviations, and multi-dimensional central limit theorems for costs on path sets, under natural hypotheses (G1)–(G3) and their translation analogues (T1)–(T3). A central contribution is a method to center fluctuations via reparameterized transfer operators, derive asymptotics from a two-variable complex function , and obtain non-degenerate Gaussian limits with explicit covariance structures. The results extend to holonomy vectors and visit counts to singularities, and a positive variance criterion ensures the CLTs are non-degenerate; this provides a rigorous probabilistic understanding of how geometric length and saddle-connection combinatorics relate on flat surfaces, with potential implications for quantitative geometry and Teichmüller dynamics.

Abstract

We study counting limit laws that compare length functions on infinite graphs. We then apply these results to flat surfaces to obtain a statistical comparison between the geometric length and the number of singularities visited by geodesic paths.

Paper Structure

This paper contains 16 sections, 26 theorems, 137 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Translation surfaces
  3. Method and ideas behind the proof
  4. Infinite directed graphs
  5. Counting for graphs
  6. Preliminaries
  7. The matrices $W_{s,t}$
  8. Complex functions
  9. Law of large numbers
  10. Reparameterising the matrices
  11. Large deviations
  12. Central limit theorems
  13. Deducing the Central Limit Theorem
  14. Translation surfaces
  15. Proof of main results
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $(X,\omega)$ be a translation surface and fix a singularity $x \in\mathcal{X}$. Let $|\cdot|$ and $\ell(\cdot)$ denote the singularity and geometric length of a saddle connection path respectively. Then there exists $\Lambda > 0$ such that for any $\epsilon > 0$ Furthermore there exists $\sigma^2 > 0$ such that for any $a,b \in {\mathbb R}, a<b$

Figures (2)

  • Figure 1: A simple way to present a translation surface is by identifying opposite sides of a polygon in the plane. On the left is an L-shaped translation surface with one singularity (indicated by the dots). The picture on the right shows a saddle connection path $p = (s_1,s_2,s_3)$ on the surface. The angle between $s_1$ and $s_2$ is $5\pi/4$ and the angle between $s_2$ and $s_3$ is $13\pi/6$.
  • Figure 2: A translation surface with $4$ singularities.

Theorems & Definitions (65)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Remark 2.1
  • Definition 2.2
  • Example 2.3
  • ...and 55 more