Introduction: Motion on Lattices

The scope of these notes is to present a model case –but already complex enough– of motion in heterogeneous media. Even though this analysis will be performed in the relatively simplified setting of a periodic lattice, where the heterogeneous structure is somewhat built-in in the environment itself, this must be thought of as a case study for a large class of inhomogeneous media. In such a lattice setting we consider the simplest order parameter –obtained by labelling the nodes of the lattice with zeros or ones–, and define an energy that favours constant values of such an order parameter and penalizes the creation of ‘discrete interfaces’.