Localization of the Maximal Entropy Random Walk

We define a new class of random walk processes which maximize entropy. This maximal entropy random walk is equivalent to generic random walk if it takes place on a regular lattice, but it is not if the underlying lattice is irregular. In particular, we consider a lattice with weak dilution. We show that the stationary probability of finding a particle performing maximal entropy random walk localizes in the largest nearly spherical region of the lattice which is free of defects. This localization phenomenon, which is purely classical in nature, is explained in terms of the Lifshitz states of a certain random operator.

Figure 1

Density plots of ${\pi }_i^{*}$ for a periodic square lattice of size $L\times L$, for $L=40$ and the fractions $q=0.001$, 0.01, 0.1 of removed links. The nodes incident with removed links are marked with circles. Data are obtained numerically by applying the Lanczos algorithm to the adjacency matrix. For a sparse matrix used here, the algorithm has a complexity $O(N^2)$, where $N=L^2$ in two dimensions. Plots for other values $L,q$ can be produced using the demonstration [16].

Figure 2

Top: a ladder with randomly removed rungs. Bottom: stationary distribution ${\pi }_i^{*}$ on the ladder for $L=500$ and five randomly positioned defects marked with vertical lines.

Figure 3

Ground-state energy $E_0$ of Eq. (13) on ladders versus $\ln L$ for $L=20,\dots ,960$ and $q=0.1$. The solid line shows the estimate $E_0^{-1/2}=\ln L/(\pi |\ln p|)+B$, with $B$ fitted to the rightmost data point.