Volumes for $${\mathrm{SL}}_N({\mathbb {R}})$$ SL N ( R ) , the Selberg Integral and Random 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 that the singular values lie between $$R_1$$ and $$1/R_2$$ in the former, and that the 2-norm, or alternatively the Frobenius norm, is bounded by R in the latter. By a result of Duke, Rudnick 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.