Finite-size effects and percolation properties of Poisson geometries

Random tessellations of the space represent a class of prototype models of heterogeneous media, which are central in several applications in physics, engineering, and life sciences. In this work, we investigate the statistical properties of $d$-dimensional isotropic Poisson geometries by resorting to Monte Carlo simulation, with special emphasis on the case $d=3$. We first analyze the behavior of the key features of these stochastic geometries as a function of the dimension $d$ and the linear size $L$ of the domain. Then, we consider the case of Poisson binary mixtures, where the polyhedra are assigned two labels with complementary probabilities. For this latter class of random geometries, we numerically characterize the percolation threshold, the strength of the percolating cluster, and the average cluster size.

Figure 1

Cutting a cube with a random plane. A cube of side $L$ is centered in $O$. The circumscribed sphere centered in $O$ has a radius $R=\sqrt{3}L/2$. The point $\mathbf{M}$ is defined by $\mathbf{M}=r\mathbf{n}$, where $r$ is uniformly sampled in the interval $[0,R]$ and $\mathbf{n}$ is a random unit vector of components $\mathbf{n}={(n_1,n_2,n_3)}^T$, with $n_1=1-2{\xi }_1, n_2=\sqrt{1-n_1^2}cos\left(2\pi {\xi }_2\right)$ and $n_3=\sqrt{1-n_1^2}sin\left(2\pi {\xi }_2\right)$. The auxiliary variables ${\xi }_1$ and ${\xi }_2$ are sampled from two independent uniform distributions in the interval $[0,1]$. The random plane $K$ of equation $n_{1}x+n_{2}y+n_{3}z=r$ is orthogonal to the vector $\mathbf{n}$ and intersects the point $\mathbf{M}$.

Figure 2

Examples of Monte Carlo realizations of isotropic Poisson geometries restricted to a three-dimensional box of linear size $L$. For all realizations, we have chosen $L=20$. The geometry at (a) the top has $\lambda =0.2$, that at (b) the middle has $\lambda =1$ and that at (c) the bottom has $\lambda =3$. For fixed $L$, the average number of random polyhedra composing the geometry increases with increasing $\lambda$.

Figure 3

The average number $\langle N_p|L\rangle$ of $d$ polyhedra in $d$-dimensional Poisson geometries as a function of the linear size $L$ of the domain. The scaling law $L^d$ is displayed for reference with dashed lines. Inset. The dispersion factor $\sigma \left[N_p\right|L]/\langle N_p|L\rangle$ as a function of $L$. The scaling law $1/\sqrt{L}$ is displayed for reference with dashed lines.

Figure 4

The probability densities $\mathcal{P}\left(\ell \right|L)$ of the segment lengths as a function of the linear size $L$ of the domain, in dimension $d=3$. Symbols correspond to the Monte Carlo simulation results, with lines added to guide the eye: blue triangles denote $L=1$, red diamonds $L=2$, green circles $L=3$, black squares $L=5$, and purple crosses $L=40$. The asymptotic (i.e., $L\to \infty$) exponential distribution given in Eq. (1) is displayed as a black dashed line for reference. The inset displays the same data in log-linear scale.

Figure 5

The probability densities $\mathcal{P}\left(\ell \right|L)$ of the segment lengths as a function of the linear size $L$ of the domain and of the dimension $d$. Symbols correspond to the Monte Carlo simulation results, with lines added to guide the eye: for $L=1$, blue crosses denote $d=1$, green circles $d=2$, and orange triangles $d=3$; for $L=2$, red pluses denote $d=1$, gray squares $d=2$, and purple diamonds $d=3$. The asymptotic (i.e., $L\to \infty$) exponential distribution given in Eq. (1) is displayed as a black dashed line for reference. Inset. The case of a system size $L=40$: red crosses denote $d=1$, green circles $d=2$, and blue diamonds $d=3$; the black dashed line corresponds to Eq. (1).

Figure 6

The probability density $\mathcal{P}\left(r_{in}\right|L)$ of the inradius as a function of the system size $L$ and of the dimension $d$. Symbols correspond to Monte Carlo simulation results. For $d=1$, red squares denote $L=5$ and blue pluses $L=40$. For $d=2$, red crosses denote $L=5$ and blue diamonds $L=40$. For $d=3$, red circles denote $L=5$ and blue triangles $L=40$. The black dashed lines represent the asymptotic (i.e., $L\to \infty$) distribution in Eq. (7). Inset. Comparison between $\mathcal{P}\left(r_{in}\right|L)$ for a typical polyhedron (blue triangles) and ${\mathcal{P}}_0\left(r_{in}\right|L)$ for the polyhedron containing the origin (green circles), for $d=3$ and $L=40$. The dashed line represents the asymptotic distribution in Eq. (7).

Figure 7

The probability density $\mathcal{P}\left(Y_d\right|L)$ of the dimensionless $d$-volume $Y_d=V_d/\langle V_d\rangle$ as a function of the linear size $L$ of the domain and of the dimension $d$. Black inverted triangles denote a system size $L=40$ for $d=1$. For $d=2$, purple diamonds are chosen for a system size $L=40$ and orange squares for $L=10$. For $d=3$, blue crosses are chosen for a system size $L=40$, red circles for $L=10$, and gray triangles for $L=5$. For $d=1$, the asymptotic (i.e., $L\to \infty$) exponential distribution given in Eq. (1) is displayed as a black dashed line. For $d=2$ and $d=3$, dashed lines denote exponential decay. Inset. Comparison between $\mathcal{P}\left(V_d\right|L)$ for a typical polyhedron (blue triangles) and ${\mathcal{P}}_0\left(V_d\right|L)$ for the polyhedron containing the origin (green circles), for $d=3$.

Figure 8

The dimensionless first moment $\langle Y_d^1|L\rangle =\langle V_d^1|L\rangle /\langle V_d^1\rangle$ of the $d$ volume, as a function of the system size $L$ and of the dimension $d$. Monte Carlo simulation results are displayed as symbols, with dashed lines lines to guide the eye for $d=2$ and $d=3$. For $d=1$, the solid line represents the exact formula given in Eq. (5). Red diamonds denote $d=1$; green circles denote $d=2$; blue triangles denote $d=3$. Inset. The dimensionless moments $\langle Y_d^m|L\rangle =\langle V_d^m|L\rangle /\langle V_d^m\rangle$ of the $d$ volume, for $m=1,2,3$, as a function of the system size $L$, for $d=1$ and $d=2$. Monte Carlo simulation results are displayed as symbols, with dashed lines lines to guide the eye for $d=3$. For $d=1$, the solid line represents the exact formula given in Eq. (5). For $d=1$, red diamonds denote $m=1$; red pluses denote $m=2$; red inverted triangles denote $m=3$. For $d=3$, blue triangles denote $m=1$; blue crosses denote $m=2$; blue squares denote $m=3$.

Figure 9

Examples of Monte Carlo realizations of colored isotropic Poisson geometries restricted to a three-dimensional box of linear size $L$. For all realizations, we have chosen $L=20$. The geometry at (a) the top has $\lambda =0.3$ and $p=0.5$; the geometry in (b) the center has $\lambda =1$ and $p=0.5$; the geometry at (c) the bottom has $\lambda =1$ and $p=0.25$.

Figure 10

Monte Carlo simulation of the percolation probability ${\mathcal{P}}_C\left(p\right|L)$ for $d=3$ as a function of the coloring probability $p$ and of the system size $L$. Purple crosses represent $L=30$, green diamonds $L=40$, orange squares $L=60$, blue triangles $L=80$, and red circles $L=100$. Curves have been added to guide the eye. The estimated $p_c$ is displayed as a dashed line, with confidence error bars drawn as thinner dashed lines. For all sizes $L$ we have generated ${10}^3$ realizations, with the exception of $L=100$, for which $5\times {10}^2$ realizations were generated.

Figure 11

Monte Carlo simulation of the segment length distributions ${\mathcal{P}}_r\left(\ell \right|L)$ and ${\mathcal{P}}_r^{†}\left(\ell \right|L)$ for $d=3$ as a function of the coloring probability $p$. Purple crosses represent ${\mathcal{P}}_r\left(\ell \right|L)$ with $p=0.2$; blue triangles: ${\mathcal{P}}_r\left(\ell \right|L)$ with $p=0.6$; red diamonds: ${\mathcal{P}}_r\left(\ell \right|L)$ with $p=0.8$. Green circles denote the segment length distribution ${\mathcal{P}}_r^{†}\left(\ell \right|L)$ for $p=0.2$. All simulations have been performed for a system size $L=40$ and $5\times {10}^3$ realizations. For each $p$, the black dashed lines correspond to the exponential distribution ${\mathcal{P}}_r\left(\ell \right|L)$ given in Eq. (13).

Figure 12

Monte Carlo simulation of the segment length distributions ${\mathcal{P}}_r\left(\ell \right)$ and ${\mathcal{P}}_r^{†}\left(\ell \right)$ for $d=3$ and $L=40$ as a function of the coloring probability $p$. Gray squares denote ${\mathcal{P}}_r\left(\ell \right|L)$ for $p=0.6$ and red circles denote ${\mathcal{P}}_r\left(\ell \right|L)$ for $p=0.8$. Purple crosses denote ${\mathcal{P}}_r^{†}\left(\ell \right)$ for $p=0.6$, green diamonds ${\mathcal{P}}_r^{†}\left(\ell \right)$ for $p=0.8$, blue triangles ${\mathcal{P}}_r^{†}\left(\ell \right|L)$ for $p=0.95$. The dashed curve corresponds to the chord length distribution $h_I\left(\ell \right|L)$ of a cube, as given in Eq. (14). Inset: Effects of system size $L$ for fixed $p=0.8$. Black squares denote ${\mathcal{P}}_r\left(\ell \right|L)$ for $L=20$; red circles denote ${\mathcal{P}}_r\left(\ell \right|L)$ for $L=40$. Orange triangles denote ${\mathcal{P}}_r^{†}\left(\ell \right|L)$ for $L=20$; green diamonds denote ${\mathcal{P}}_r^{†}\left(\ell \right|L)$ for $L=40$. The chord length distribution $h_I\left(z\right), z=\ell /L$, is displayed as a dotted curve for $L=20$; and as a dashed curve for $L=40$.

Figure 13

The average cluster size $S\left(p\right|L)$ as a function of the coloring probability $p$ and of the system size $L$. Purple crosses represent $L=10$, green diamonds $L=20$, orange squares $L=30$, blue triangles $L=40$, and red circles $L=60$. Curves have been added to guide the eye. The estimated $p_c$ is displayed as a dashed line for reference. For all sizes $L$ we have generated ${10}^3$ realizations. Inset: The behavior of $S\left(p\right|L)$ as a function of $p-p_c^{*}$, where $p_c^{*}$ is our best estimate for the percolation threshold, namely, $p_c^{*}=0.290$. Blue triangles correspond to $L=40$ and red circles to $L=60$. The dashed line corresponds to the power-law scaling $S\left(p\right)\propto |p-p_c{|}^{-\gamma }$, with $\gamma =1.793$.

Figure 14

The percolation strength $P\left(p\right|L)$ as a function of the coloring probability $p$ and of the system size $L$. Purple crosses represent $L=10$, green diamonds $L=20$, orange squares $L=30$, blue triangles $L=40$, and red circles $L=60$. Curves have been added to guide the eye. The estimated $p_c$ is displayed as a solid line for reference. For all sizes $L$ we have generated ${10}^3$ realizations. Inset: The behavior of $P\left(p\right|L)$ as a function of $p-p_c^{*}$, where $p_c^{*}$ is our best estimate for the percolation threshold, namely, $p_c^{*}=0.290$. Blue triangles correspond to $L=40$ and red circles to $L=60$. The dashed line corresponds to the power-law scaling $P\left(p\right)\propto {(p-p_c)}^{\beta }$, with $\beta =0.4181$.