  Polygon spaces like $M_\ell=\{(u_1,...,u_n)\in S^1\times... S^1 ;\
\sum_{i=1}^n l_iu_i=0\}/SO(2)$ or they three dimensional analogues $N_\ell$
play an important r\^ole in geometry and topology, and are also of interest in
robotics where the $l_i$ model the lengths of robot arms. When $n$ is large,
one can assume that each $l_i$ is a positive real valued random variable,
leading to a random manifold. The complexity of such manifolds can be
approached by computing Betti numbers, the Euler characteristics, or the
related Poincar\'e polynomial. We study the average values of Betti numbers of
dimension $p_n$ when $p_n\to\infty$ as $n\to\infty$. We also focus on the
limiting mean Poincar\'e polynomial, in two and three dimensions. We show that
in two dimensions, the mean total Betti number behaves as the total Betti
number associated with the equilateral manifold where $l_i\equiv \bar l$. In
three dimensions, these two quantities are not any more asymptotically
equivalent. We also provide asymptotics for the Poincar\'e polynomials
