  We review various features of the statistics of random paths on graphs. The
relationship between path statistics and Quantum Mechanics (QM) leads to two
canonical ways of defining random walk on a graph, which have different
statistics and hence different entropies. Generic random walk (GRW) is in
correspondence with the field-theoretical formalism, whereas maximal entropy
random walk (MERW), introduced by us in a recent work, is motivated by the
Feynman path-integral formulation of QM. GRW maximizes entropy locally
(neighbors are chosen with equal probabilities), in contrast to MERW which does
so globally (all paths of given length and endpoints are equally probable). The
stationary distribution for MERW is given by the ground state of a
quantum-mechanical problem where nodes whose degree is smaller than average act
as repulsive impurities. We investigate static and dynamical properties GRW and
MERW in a variety of examples in one and two dimensions. The most spectacular
difference arises in the case of weakly diluted lattices, where a particle
performing MERW gets eventually trapped in the largest nearly spherical region
which is free of impurities. We put forward a quantitative explanation of this
localization effect in terms of a classical Lifshitz phenomenon.
