Volumes for ${\rm SL}_N(\mathbb R)$, the Selberg integral and random lattices

TL;DR

This paper computes asymptotic volumes for SL_N(R) and GL_N(R) matrices under norm bounds, relates these to lattice point counting, and explores sampling and distribution of shortest vectors in random lattices for small N.

Contribution

It provides explicit asymptotic volume formulas for matrix groups with norm restrictions and connects these to lattice point counting and probabilistic distributions of shortest vectors.

Findings

Asymptotic volume formulas for SL_N(R) and GL_N(R) under norm bounds.

Connection between volume asymptotics and lattice point counting in SL_N(Z).

Explicit distributions for shortest vectors in 2- and 3-dimensional lattices.

Abstract

There is a natural left and right invariant Haar measure associated with the matrix groups GL${}_N(\mathbb R)$ and SL${}_N(\mathbb R)$ due to Siegel. For the associated volume to be finite it is necessary to truncate the groups by imposing a bound on the norm, or in the case of SL${}_N(\mathbb R)$, by restricting to a fundamental domain. We compute the asymptotic volumes associated with the Haar measure for GL${}_N(\mathbb R)$ and SL${}_N(\mathbb R)$ matrices in the case of that the operator norm lies between $R_1$ and $1/R_2$ in the former, and this norm, or alternatively the 2-norm, is bounded by $R$ in the latter. By a result of Duke, Rundnick and Sarnak, such asymptotic formulas in the case of SL${}_N(\mathbb R)$ imply an asymptotic counting formula for matrices in SL${}_N(\mathbb Z)$. We discuss too the sampling of SL${}_N(\mathbb R)$ matrices from the truncated sets. By then using lattice reduction to a fundamental domain, we obtain histograms approximating the probability density functions of the lengths and pairwise angles of shortest length bases vectors in the case $N=2$ and 3, or equivalently of shortest linearly independent vectors in the corresponding random lattice. In the case $N=2$ these distributions are evaluated explicitly.