  How much can randomness help computation? Motivated by this general question
and by volume computation, one of the few instances where randomness provably
helps, we analyze a notion of dispersion and connect it to asymptotic convex
geometry. We obtain a nearly quadratic lower bound on the complexity of
randomized volume algorithms for convex bodies in R^n (the current best
algorithm has complexity roughly n^4, conjectured to be n^3). Our main tools,
dispersion of random determinants and dispersion of the length of a random
point from a convex body, are of independent interest and applicable more
generally; in particular, the latter is closely related to the variance
hypothesis from convex geometry. This geometric dispersion also leads to lower
bounds for matrix problems and property testing.
