The behavior of random reduced bases

Abstract

We prove that the number of Siegel-reduced bases for a randomly chosen $n$-dimensional lattice becomes, for $n \rightarrow \infty$, tightly concentrated around its mean. We also show that most reduced bases behave as in the worst-case analysis of lattice reduction. Comparing with experiment, these results suggest that most reduced bases will, in fact, "very rarely" occur as an output of lattice reduction. The concentration result is based on an analysis of the spectral theory of Eisenstein series and uses (probably in a removable way) the Riemann hypothesis.