  Inspired by a remarkable work of F\'{e}lix, Halperin and Thomas on the
asymptotic estimation of the ranks of rational homotopy groups, and more recent
works of Wu and the authors on local hyperbolicity, we prove two asymptotic
formulae for torsion rank of homotopy groups, one using ordinary homology and
one using $K$-theory. We use these to obtain explicit quantitative asymptotic
lower bounds on the torsion rank of the homotopy groups for many interesting
spaces after suspension, including Moore spaces, Eilenberg-MacLane spaces,
complex projective spaces, complex Grassmannians, Milnor hypersurfaces and
unitary groups.
