Minimal pseudo-Anosov translation lengths on the complex of curves

We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus $g \geqslant 2$, we show that there are positive constants $a_1 < a_2$ such that the minimal translation length is bounded below and above by $a_1/ g^2$ and $a_2/g^2$.