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

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 self-similar 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 $n\mathrm{th}$ generation graph of the family, this probability is shown to be given by the $n\mathrm{th}$ iterate of a certain scaling function or map $q\to f\left(q\right)$. 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 two and three 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.

Figure 1

Behavior of $P_j\left(q\right)$ for $N=4$.

Figure 2

Behavior of $T_j\left(q\right)$ for $N=4$.

Figure 3

Sierpinski gasket with $N\equiv N_2=15$.

Figure 4

Sierpinski tower with $N=10$.

Figure 5

Mean walk length versus survival probability $q$ for the $N=15$ Sierpinski gasket (traps at two corner sites).

Figure 6

Mean walk length versus survival probability $q$ for the $N=42$ Sierpinski gasket (traps at two corner sites).

Figure 7

Mean walk length versus survival probability $q$ for the $N=10$ Sierpinski tower (traps at three corner sites).

Figure 8

Mean walk length versus survival probability $q$ for the $N=34$ Sierpinski tower (traps at three corner sites).