  We study the trapping phenomenon of random walks in random environments of
i.i.d. non-negative random conductances on the bonds of the grid
$\mathbb{Z}^d$, the so-called Random Conductance Model. For the important case
of conductances in $[0, 1]$ with polynomial-tailed law near zero, we provide an
intriguing improvement of the anomalous return probability lower bound. In a
symmetrical situation with conductances in $[1, \infty)$ with a polynomial law
at infinity, by adapting the same argument, we obtain interesting results that
also show opposite behaviors for the constant speed random walk.
