Figure 1

Modifications of the original lattice: arrows denote one-way links.Reuse & Permissions

Figure 2

(Color online) Three dimensions: Influence of the distance between the source and the target. Red diamonds: simulations; blue dashed line: evaluation of the FPT with $H=G_0$. The domain is a cube of side 41, the target being in the middle of it. All the simulation points correspond to different positions of the source.Reuse & Permissions

Figure 3

Opening in a flat reflecting boundaryReuse & Permissions

Figure 4

(Color online) Three dimensions: Relative difference between the simulations and the theoretical prevision (21). The domain is a cube of side 51 centered on the target at (0,0,0) and the source at (2,2,1). The order of magnitude of the relative difference is indeed $n{N}^{-2∕3}$.Reuse & Permissions

Figure 5

(Color online) Simulation of the probability distribution of the FPT, rescaled as a probability density, with different domain sizes. In both cases, the target is in the middle of a cube at position (0,0,0) and the source is in (1,2,2). We plot the estimated density (22) vs numerical simulations for different domain sizes. The dark blue (simulation) and cyan (estimated density) curves correspond to a cube of side 11 $(N=1331)$; the red (simulation) and orange (estimated density) correspond to a cube of side 21 $(N=9261)$; the green dashed curve corresponds to the high-$N$ limit (23) of the probability density. For both domain sizes, the simulated distribution splits into two parts for short times and cannot be distinguished from the theoretical density afterwards. The labels a and b correspond to domain sizes of 11 and 21, and the high-$N$ limit is labeled c.Reuse & Permissions

Figure 6

(Color online) Simulation of the distribution of the FPT in the 2D case, rescaled as a probability density. The domains are squares of different size; the target is in the middle of the squares at position (0,0) and the source is in (2,3). The side of the squares are 21 (red curve [a]), 41 (orange curve [b]), 81 (yellow curve [c]), and 161 (green curve [d]). The semilogarithmic scale shows the long-time exponential decay. The splitting of the curves for short times is due to parity effects.Reuse & Permissions

Figure 7

(Color online) Three dimensions: two-target simulations. Simulations (red crosses) vs theory with the approximation $H=G_0$ (solid line). One target is fixed in $(-5,0,0)$; the source is fixed in (5,0,0); the other target is at $(x,3,0)$. The domain is a cube of side 51, the middle is the point (0,0,0).Reuse & Permissions

Figure 8

(Color online) Three dimensions: two-target simulations. The conditions are identical to those of Fig. 7; we show the conditional absorption times $⟨{\mathbf{T}}_1⟩$ (respectively, $⟨{\mathbf{T}}_2⟩$). The blue crosses (respectively, red pluses) show the results of the numerical simulations, the cyan [a] (respectively, orange [b]) dashed line shows the theoretical expression (32) with $H=G_0$, the green [c] (respectively, brown [d]) solid line shows the theoretical expression with the exact value of $H$ [Eq. (A3)].Reuse & Permissions

Figure 9

(Color online) Continuous problem.Reuse & Permissions

Figure 10

(Color online) Brownian motion on a 2D disk of radius 25 centered on (0,0); the source is in (0,1) and the target of radius 1 is on $(x,0)$. Red crosses: simulations; black solid line: estimation (49) with the exact function $H$ for a sphere given by Eq. (A6).Reuse & Permissions

Figure 11

(Color online) Brownian motion on a 2D disk of radius 25 centered on (0,0); $D=1$; the source is in $(-5,2)$ and the two targets of radius 1 are on (5,2) $\left(T_1\right)$ and $(x,0)$ $\left(T_2\right)$. Red crosses: simulations; black solid line: estimations (63, 64) with the exact function $H$ for a disk (A6).Reuse & Permissions

Figure 12

(Color online) Brownian motion on a 3D sphere of radius 25 centered on (0,0,0); $D=1$; the source is in $(-5,2,0)$ and the two targets are on (5,2,0) ($T_1$, of radius 0.5) and $(x,0,0)$ ($T_2$, of radius 1.5). Red crosses: simulations; black solid line: estimations (63, 64), with the exact function $H$ for a sphere (A7).Reuse & Permissions

Figure 13

(Color online) Two-target simulations. The conditions are identical to those of Fig. 12; we show the conditional absorption times $⟨{\mathbf{T}}_1⟩$ (respectively, $⟨{\mathbf{T}}_2⟩$). The blue $Xs$ (respectively, red $+s$) show the results of numerical simulations; The green [a] (respectively, brown [b]) solid line shows the theoretical estimation (69) with the exact function $H$ for a sphere (A7).Reuse & Permissions

Figure 14

(Color online) Schematic picture of the quantities used in the computation of $H(\mathbf{r}\mid {\mathbf{r}}^{\prime })$.Reuse & Permissions

Figure 15

(Color online) Discrete random walk: Influence of the position of the source; the domain is a 2D square of side 41 centered on (0,0,0), the target is at $(-18,0)$, and the source is at $(x-20,0)$; blue dashed line: approximation $H=G_0$; black solid line: local approximation taking into account the boundary.Reuse & Permissions

Figure 16

(Color online) 3D Brownian motion: the domain is an eight of sphere; the sphere is of radius 25, centered on (0,0,0), and the domain is reduced to positive coordinates. The target is in (10,10,2), and the source is in $(x,x,3)$. Red crosses: numerical simulations; blue dashed curve: basic approximation $H=G_0$; solid black curve: approximation (C7) with $H$ taking into account the nearest boundary [Eq. (A13)].Reuse & Permissions

Figure 17

(Color online) Picture of the real and image charges when the target is near the boundary ($+=\text{red}$ pluses; $-=\text{blue}$ crosses).Reuse & Permissions