Note on Return Probability Behaviors in Random Environments of   Conductances
  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. 
Omar Boukhadra
math.PR