We consider laws of the iterated logarithm and the rate function for sample paths of random walks on random conductance models under the assumption that the random walks enjoy long time sub-Gaussian heat kernel estimates.
"Rate functions for random walks on random conductance models and related topics." Kodai Math. J. 40 (2) 289 - 321, June 2017. https://doi.org/10.2996/kmj/1499846599