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Betti numbers of configuration spaces of mechanical linkages (known
also as polygon spaces) depend on a large number of parameters –
the lengths of the bars of the linkage. Motivated by applications in
topological robotics, statistical shape theory and molecular biology,
we view these lengths as random variables and study asymptotic values of
the average Betti numbers as the number of links n tends to infinity.
We establish a surprising fact that for a reasonably ample class of
sequences of probability measures the asymptotic values of the average
Betti numbers are independent of the choice of the measure. The main
results of the paper apply to planar linkages as well as for linkages
in R3. We also prove results about higher moments of Betti
numbers.
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