Sum Over Histories: Discrete Step Interpretation
  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. 
Muhammad Adeel Ajaib
quant-ph