we study on-diagonal heat kernel estimates and exit time estimates for continuous time random walks (ctrws) among i.i.d. random conductances with a power-law tail near zero. for two types of natural ctrws, we give optimal exponents of the tail such that the behaviors are standard (i.e. similar to the random walk on the euclidean space) above the exponents. we then establish the local clt for the ctrws. we will also compare our results to the recent results by andres-deuschel-slowik. this talk is a joint work with o. boukhadra (constantine) and p. mathieu (marseille).