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arXiv · 2510.11371

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

Jens Marklof·Andreas Strömbergsson·Shucheng Yu
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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.

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Jens Marklof, Andreas Strömbergsson, Shucheng Yu. 2025-10-13. Extreme events and impact statistics for unipotent actions on the space of lattices. https://arxiv.org/abs/2510.11371

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