Global first-passage times of fractal lattices

The global first passage time density of a network is the probability that a random walker released at a random site arrives at an absorbing trap at time $T$. We find simple expressions for the mean global first passage time $⟨T⟩$ for five fractals: the $d$-dimensional Sierpinski gasket, $T$ fractal, hierarchical percolation model, Mandelbrot-Given curve, and a deterministic tree. We also find an exact expression for the second moment $⟨T^2⟩$ and show that the variance of the first passage time, $\mathrm{Var}\left(T\right)$, scales with the number of nodes within the fractal $N$ such that $\mathrm{Var}\left(T\right)\sim N^{4∕\overline{d}}$, where $\overline{d}$ is the spectral dimension.

Figure 1

(a) First iteration of the Sierpinski Gasket lattice. (b) Zeroth iteration of the Sierpinski gasket lattice. The labeling used in the above (a) and (b) implies that the left corner of the $n\text{th}$ iteration of the 2D Sierpinski gasket lattice is denoted by $3(3^n+1)∕2-2^n$, the right corner by $3(3^n+1)∕2$ and the apex by 1.

Figure 2

Illustration of the method of images argument used to prove the result ${\alpha }_{k,n}=p_{1k,n-1}$. The filled circles denote sources and the open circles are the concentrations of interest.

Figure 3

Deterministic tree (a) $n=1$ (b) $n=0$. The labeling of the nodes 1 to 5 remain the same regardless of the iteration.

Figure 4

The $T$ tree (a) $n=1$, (b) $n=0$. The labeling of the nodes remains the same for all iterations.

Figure 5

Mandelbrot-Given curve (a) $n=1$, (b) $n=0$.

Figure 6

Hierarchicial percolation model (a) $n=1$, (b) $n=0$.

Figure 7

The normalized probability for a random walker to arrive at a trap at node 1 on the $m\text{th}$ jump for the 10th iteration of the Sierpinski gasket $(d=2)$ (○), Sierpinski gasket $(d=3)$ (+), and the hierarchical percolation model (◇). The values are calculated from the Taylor series of the GFPT generating function $f_1^{*}\left[\ln \left(z\right)\right]$. A symbolic algebra package was used.