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Exact solution for statics and dynamics of maximal-entropy random walks on Cayley trees

J. K. Ochab^* and Z. Burda^†

Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagiellonian University, Reymonta 4, PL-30-059 Kraków, Poland

^*jeremi.ochab@uj.edu.pl
^†zdzislaw.burda@uj.edu.pl

Phys. Rev. E 85, 021145 – Published 24 February, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.021145

Abstract

We provide analytical solutions for two types of random walk: generic random walk (GRW) and maximal-entropy random walk (MERW) on a Cayley tree with arbitrary branching number, root degree, and number of generations. For MERW, we obtain the stationary state given by the squared elements of the eigenvector associated with the largest eigenvalue $λ_{0}$ of the adjacency matrix. We discuss the dynamics, depending on the second largest eigenvalue $λ_{1}$ , of the probability distribution approaching to the stationary state. We find different scaling of the relaxation time with the system size, which is generically shorter for MERW than for GRW. We also signal that depending on the initial conditions, there are relaxations associated with lower eigenvalues which are induced by symmetries of the tree. In general, we find that there are three regimes of a tree structure resulting in different statics and dynamics of MERW; these correspond to strongly, critically, and weakly branched roots.

Article Text

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