Hitting and trapping times on branched structures

In this work we consider a simple random walk embedded in a generic branched structure and we find a close-form formula to calculate the hitting time $H\left(i,f\right)$ between two arbitrary nodes $i$ and $j$. We then use this formula to obtain the set of hitting times $\left\{H\left(i,f\right)\right\}$ for combs and their expectation values, namely, the mean first-passage time, where the average is performed over the initial node while the final node $f$ is given, and the global mean first-passage time, where the average is performed over both the initial and the final node. Finally, we discuss applications in the context of reaction-diffusion problems.

Figure 1

Example of the branched structure $\mathcal{V}$ obtained by joining to each point $j$ of a graph ${\mathcal{G}}_0$, called the base (left), a graph ${\mathcal{G}}_j$, called the fiber (middle).

Figure 2

Examples of two-dimensional combs, where the backbone is endowed with reflecting boundary conditions (top) or periodic boundary conditions (bottom). The size of the backbone is referred to as $2L+1$, while the length of the teeth $L_y$ is taken to be uniform and to scale linearly with $L$, namely, $L_y=\alpha L$. For instance, at the top $L=7$ and $\alpha L=5$.

Figure 3

Schematization of the decomposition used in Eq. (2), where $\mathcal{V}=\mathcal{I}\cup \mathcal{F}\cup \mathcal{B}$. Note that the whole set of fiber graphs $\left\{{\mathcal{G}}_k\right\}$ also includes the set of nodes making up the base graph because the joining nodes, also called “roots,” are shared.

Figure 4

Example of a Sierpinski gasket of generation $n=3$, branched with linear chains with $L=5$ vertices.

Figure 5

Example of a three-dimensional comb. Vertices which belong to the first generation are shown in green; those which belong to the second, in lilac; and those which belong to the third, in red.

Figure 6

Numerical check of the theoretical predictions of Eq. (17) for $\{x_f,y_f\}=\{0,0\}$. In both panels the results of the simulations are divided by the theoretical value of ${\text{MFPT}}_{\{0,0\}}$.

Figure 7

Numerical check of the theoretical predictions of Eq. (17) for $\{x_f,y_f\}=\{0,\alpha L\}$. In both panels the results of the simulations are divided by the theoretical value of ${\text{MFPT}}_{\{0,\alpha L\}}$.

Figure 8

Numerical check of the theoretical predictions of Eq. (17) for $\{x_f,y_f\}=\{L,\alpha L\}$. In both panels the results of the simulations are divided by the theoretical value of ${\text{MFPT}}_{\{L,\alpha L\}}$.

Figure 9

Numerical check of the theoretical predictions of Eq. (21) for $y_f=0$. In both panels the results of the simulations are divided by the theoretical value of ${\text{MFPT}}_{\left(0\right)}^{⥀}$.

Figure 10

Numerical check of the theoretical predictions of Eq. (21) for $y_f=\alpha L$. In both panels the results of the simulations are divided by the theoretical value of ${\text{MFPT}}_{\alpha L}^{⥀}$.

Figure 11

Numerical check of the theoretical predictions of Eq. (27). In both panels the results of the simulations are divided by the theoretical value of $\text{GMFPT}$.

Figure 12

Numerical check of the theoretical predictions of Eq. (29). In both panels the results of the simulations are divided by the theoretical value of ${\text{GMFPT}}^{⥀}$.

Figure 13

Phase diagram of ${\text{MFPT}}_{\{X,Y\}}-\text{GMFPT}$: vertices where ${\text{MFPT}}_{\{X,Y\}}>\text{GMFPT}$ are shown in yellow; those where ${\text{MFPT}}_{\{X,Y\}}<\text{GMFPT}$, in teal.

Figure 14

Phase diagram of ${\text{MFPT}}_{\left(Y\right)}^{⥀}-{\text{GMFPT}}^{⥀}$. Vertices where ${\text{MFPT}}_{\left(Y\right)}>\text{GMFPT}$ are shown in yellow; those where ${\text{MFPT}}_{\left(Y\right)}<\text{GMFPT}$, are in teal.

Figure 15

Each curve is associated with a different value of ${\text{MFPT}}_f$ (as reported in the pertinent box) and represents the set of positions $(X,Y)$ for A compatible with that value of the MFPT. Here we show results for $\alpha =10$ (top) and for $\alpha =1/10$ (bottom).