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We provide analytical solutions for two types of random walk: generic randomwalk (GRW) and maximal entropy random walk (MERW) on a Cayley tree witharbitrary branching number, root degree, and number of generations. For MERW,we obtain the stationary state given by the squared elements of the eigenvectorassociated with the largest eigenvalue $\lambda_0$ of the adjacency matrix. Wediscuss the dynamics, depending on the second largest eigenvalue $\lambda_1$,of the probability distribution approaching to the stationary state. We finddifferent scaling of the relaxation time with the system size, which isgenerically shorter for MERW than for GRW. We also signal that depending on theinitial conditions there are relaxations associated with lower eigenvalueswhich are induced by symmetries of the tree. In general, we find that there arethree regimes of a tree structure resulting in different statics and dynamicsof MERW; these correspond to strongly, critically, and weakly branched roots.

Exact Solution for Statics and Dynamics of Maximal Entropy Random Walk on Cayley Trees