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

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.

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