The various facets of random walk entropy

TL;DR

This paper compares two types of random walks on graphs—generic and maximal entropy—highlighting their differences in entropy, stationary distribution, and localization effects, with implications for understanding quantum-inspired path statistics.

Contribution

We introduce and analyze the maximal entropy random walk (MERW), contrasting it with the traditional generic random walk (GRW), and explore their distinct properties and localization phenomena.

Findings

MERW maximizes entropy globally, leading to different stationary distributions.

MERW causes localization in weakly diluted lattices, trapping particles in impurity-free regions.

The stationary distribution of MERW relates to the quantum ground state.

Abstract

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.