On Maximal Displacement of Bridges in the Random Conductance model

We study a discrete time random walk in an environment of i.i.d. non-negative conductances in $mathbb{Z}^d$. We consider the maximum displacements for bridges, i.e. we condition the random walk on returning to the origin, and we prove first a normal (diffusive) behavior under some regularity assumptions: standard heat kernel decay and polynomial volume growth. Afterwards, we prove that if the heat kernel decay is such that the return probabilities are sufficiently slow, we obtain anomalous (subdiffusive) maximal displacements for bridges.