First-passage properties of mortal random walks: Ballistic behavior, effective reduction of dimensionality, and scaling functions for hierarchical graphs

Abstract

We consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites. The configuration comprising the graph, the starting point of the walk, and the locations of the trap sites is taken to be exactly selfsimilar as one goes from one generation of the family to the next. Under these circumstances, the total probability that the walker hits a trap is determined exactly as a function of the single-step survival probability q of the mortal walker. On the n th generation graph of the family, this probability is shown to be given by the n th iterate of a certain scaling function or map q → f (q). The properties of the map then determine, in each case, the behavior of the trapping probability, the mean time to trapping, the temporal scaling factor governing the random walk dimension on the graph, and other related properties. The formalism is illustrated for the cases of a linear hierarchical lattice and the Sierpinski graphs in 2 and 3 Euclidean dimensions. We find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint.The relevance of this finding for experiments involving travel times of particles in diffusion-decay systems is discussed.

Exact and approximate mean first passage times on trees and other necklace structures: a local equilibrium approach

“…reach higher values with increasing c. As the total mean leakage amounts to c/50 (in any direction, so to and from 0, respectively), this means that the approximation becomes worse as more weight is accumulated on the non-dominating edges between the clusters. This is as expected, for the higher the total leakage, the further the graph is from having necklace structure, and the less precise the approximation equation (21).…”

“…We remark that equation (20) generalises immediately to arbitrary pairs of vertices v I , v J along the backbone and is not restricted to v 0 and v H , thus it holds that (21) for arbitrary I, J. The proof is valid for arbitrary random walkers; it implies that along the backbone of the necklace, coarse-graining according to LE preserves MFPTs.…”

“…Our proof is valid for general random walks, including irreversible random walks, as long as they converge to a steady-state distribution. In addition to this proof, which leads to equation (21), our analysis provides two main results. The first one, equation (13), is a generalisation of the popular EEL [12], which is only valid for reversible random walks; it is retrieved here as a special case of a more general equation, which does not require dynamical reversibility.…”

“…In addition, we note that for graphs with necklace structure, where (21) holds exactly, MFPTs can also be computed from equation (13). Conversely, when graphs do not have necklace structure, neither equation (13) nor equation ( 21) hold exactly, however, when deviations from the necklace structure are small, one may expect equation (21) to hold approximately. This means that MFPTs for the coarse-grained dynamics can be used as proxies for certain MFPTs in the original dynamics.…”

“…Only for very specific problems, e.g. in the presence of a high degree of symmetry and hierarchy, the problem can be solved explicitly by successive 'decimation' procedures [19][20][21]. This has led to the development of various approximation schemes-for instance, mean-field approaches based on node degree [23], or on the distance from a target [24].…”