Weighted heat kernel estimates: rate of convergence in Kolmogorov distance

This paper is concerned about random walks on random environments in the lattice ${\mathbb{Z}}^d$. This model is analyzed through ergodicity in the form of the logarithmic Sobolev inequality. We assume that the environments are random variables being independent and identically distributed. Here, we give heat kernel estimates for non-diagonal random matrices leading in dimension $d\ge 3$ a Berry-Esseen upper bound with a rate of convergence $t^{-\frac{1}{10}}$.