  We study the transition of a particle between two points such that the
particle takes discrete spatial steps in this transition. We analyze how the
sum over histories interpretation of quantum mechanics can be implemented in
this scenario. We show that the Euclidean propagator of a free particle is
recovered if the minimum space interval is of the order or greater than the De
Broglie's wavelength of the particle. We also describe the statistical
ensembles that model this transition. Furthermore, we discuss a possible
extension of this model to 2-dimensions which serves as an example to extend it
to any number of spatial dimensions.
