Betti numbers of random manifolds

Abstract

We study mathematical expectations of Betti numbers of configuration spaces of planar linkages, viewing the lengths of the bars of the linkage as random variables. Our main result gives an explicit asymptotic formulae for these mathematical expectations for two distinct probability measures describing the statistics of the length vectors when the number of links tends to infinity. In the proof we use a combination of geometric and analytic tools. The average Betti numbers are expressed in terms of volumes of intersections of a simplex with certain half-spaces.