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Extreme events and impact statistics for unipotent actions on the space of lattices

Jens Marklof, Andreas Strömbergsson, Shucheng Yu

TL;DR

The paper establishes extreme-value laws for the fluctuations of a rank-$k$ unipotent action on the space of $n$-dimensional lattices, extending the Horocycle-based results to higher-dimensional unipotent flows. It develops a robust hitting-time/impact-time framework via a shrinking surface of section, derives limit distributions for the tail of hitting times, and connects these to directional statistics of lattices and to model-set constructions in the $k<n-1$ case. A key contribution is the precise tail asymptotics for the hitting-time distribution, including small- and large-$X$ regimes, and the demonstration that these laws imply multidimensional logarithm laws for the corresponding extreme values. The work further generalizes to admissible measures and to unipotent actions conjugate to the primary flow, providing a comprehensive EVT/log-law theory for these rank-$k$ unipotent dynamics on lattices.

Abstract

This paper extends a recent extreme value law for horocycle flows on the space of two-dimensional lattices, due to Kirsebom and Mallahi-Karai, to the simplest examples of rank-$k$ unipotent actions on the space of $n$-dimensional lattices. We analyse the problem in terms of the hitting time and impact statistics for the unipotent action with respect to a shrinking surface of section, following the strategy of Pollicott and the first named author in the case of hyperbolic surfaces. If $k=n-1$, the limit law is given by directional statistics of Euclidean lattices, whilst for $k<n-1$ we observe new distributions for which we derive precise tail asymptotics.

Extreme events and impact statistics for unipotent actions on the space of lattices

TL;DR

The paper establishes extreme-value laws for the fluctuations of a rank- unipotent action on the space of -dimensional lattices, extending the Horocycle-based results to higher-dimensional unipotent flows. It develops a robust hitting-time/impact-time framework via a shrinking surface of section, derives limit distributions for the tail of hitting times, and connects these to directional statistics of lattices and to model-set constructions in the case. A key contribution is the precise tail asymptotics for the hitting-time distribution, including small- and large- regimes, and the demonstration that these laws imply multidimensional logarithm laws for the corresponding extreme values. The work further generalizes to admissible measures and to unipotent actions conjugate to the primary flow, providing a comprehensive EVT/log-law theory for these rank- unipotent dynamics on lattices.

Abstract

This paper extends a recent extreme value law for horocycle flows on the space of two-dimensional lattices, due to Kirsebom and Mallahi-Karai, to the simplest examples of rank- unipotent actions on the space of -dimensional lattices. We analyse the problem in terms of the hitting time and impact statistics for the unipotent action with respect to a shrinking surface of section, following the strategy of Pollicott and the first named author in the case of hyperbolic surfaces. If , the limit law is given by directional statistics of Euclidean lattices, whilst for we observe new distributions for which we derive precise tail asymptotics.

Paper Structure

This paper contains 17 sections, 19 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Surface of section
  3. Hitting time statistics
  4. More general measures
  5. Impact statistics
  6. Entry times to a cuspidal neighbourhood
  7. Preliminaries on entry times
  8. Proof of Corollary \ref{['cor3app']}
  9. Proof of Corollary \ref{['cor3']}
  10. Tail estimates
  11. Small $X$
  12. Large $X$
  13. Extreme value laws
  14. Multidimensional logarithm laws
  15. Conjugates
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\lambda$ be a Borel probability measure that is absolutely continuous with respect to $\mu$. Then, for any $Y\in{\mathbb R}$, where $\eta\in\operatorname{L{}}^1({\mathbb R})$ is a probability density with tail bounds The constant $\kappa>0$ in asytail0 is given by where for any $X>0$, $\operatorname{B{}}_X^k\subset {\mathbb R}^k$ and ${\mathbb B}_X^n\subset {\mathbb R}^n$ are the closed ba

Theorems & Definitions (44)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Remark 3.1
  • Definition 4.1
  • ...and 34 more