Betti numbers of random manifolds

TL;DR

This paper derives asymptotic formulas for the expected Betti numbers of configuration spaces of random planar linkages, revealing how these topological invariants behave as the number of links grows large.

Contribution

It provides explicit asymptotic expressions for the expected Betti numbers under two probability measures, combining geometric and analytic methods.

Findings

Explicit asymptotic formulas for expected Betti numbers

Connection between Betti numbers and volumes of geometric intersections

Analysis applicable to large numbers of links in planar linkages

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.