When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space, called a template bank. These may be constructed systematically as a lattice, or alternatively, by placing templates at randomly selected points in parameter space. Here, we calculate the fraction of signals lost by an $n$-dimensional random template bank (compared to a very finely spaced bank). This fraction is compared to the corresponding loss fraction for the best possible lattice-based template banks containing the same number of grid points. For dimensions $n<4$ the lattice-based template banks significantly outperform the random ones. However, remarkably, for dimensions $n>8$, the difference is negligible. In high dimensions, random template banks outperform the best known lattices.

Final published version. 9 pages, 3 figures, 1 table