Percolation in Media with Columnar Disorder

Abstract

We study a generalization of site percolation on a simple cubic lattice, where not only single sites are removed randomly, but also entire parallel columns of sites. We show that typical clusters near the percolation transition are very anisotropic, with different scaling exponents for the sizes parallel and perpendicular to the columns. Below the critical point there is a Griffiths phase where cluster size distributions and spanning probabilities in the direction parallel to the columns have power-law tails with continuously varying non-universal powers. This region is very similar to the Griffiths phase in subcritical directed percolation with frozen disorder in the preferred direction, and the proof follows essentially the same arguments as in that case. But in contrast to directed percolation in disordered media, the number of active (“growth”) sites in a growing cluster at criticality shows a power law, while the probability of a cluster to continue to grow shows logarithmic behavior.

Appendix

This appendix is devoted to provide a completely rigorous proof of bound (12). It will roughly follow the same lines as the derivation provided in Sect. 5. The argument is divided into two main steps: first we choose a ‘seed’ of area A on the plane $z=0$ composed of sites that are not touched by any removed column. Next we will show that the probability of finding a path above this seed starting from the plane $z=0$ and extending vertically up to height $z=c \exp {(c^{-2}A)}$ is bounded from below by a constant $\delta >0$, uniformly in A. Here, $c>0$ is a positive constant whose value is going to be fixed later. The important fact is to notice that the height z of the spanning path is exponentially larger than the area of the seed. Therefore, fixing a seed whose area is logarithmically small in comparison with the size of the lattice, allows us to find a path that traverses the lattice with good probability. Also the probability to find a suitable seed, which is exponentially small in A, is then a power of the lattice size. Together, these two—the probability to find a seed and the probability to find a path, given a seed—will give (12).

Let us assume from now on that the dimensions of the lattice are $L_x=L_y=L_z=L$ where L is a positive integer that we assume to be large. To start, we choose two integers $c>0$ and $n>0$ that are allowed to grow with L beyond any limit, as long as $cn \le L$. In addition we also fix a seed located in the intersection of the lattice with the plane $z=0$. We assume that this seed consists of a rectangular strip of thickness $c>0$ and length $c\log (n)$, so that $A = c^2 \log (n)$ (latter we will comment on why we chose this specific restriction for the shape of the seed). We then ask for the probability that at least one ‘good’ path exists which spans from height 0 to $z=c \exp {(c^{-2}A)} = cn \le L$.

Now denote by $\mathcal {S}_{B,z}(n)$ the event that there exists such a path $\gamma$ that satisfies:

• 1. $\gamma$ does not contain any site that has been deleted by the removal of columns;

• 2. $\gamma$ also does not contain any site that has been deleted by the Bernoulli percolation;

• 3. $\gamma$ is contained in the portion of the lattice that projects onto the seed in the plane $z=0$;

• 4. $\gamma$ starts at the seed ($z=0$) and extends up to height $z=cn$.

Similarly, we denote by $\mathcal {S}_B (n)$ the event that there exists a path satisfying conditions 2 to 4, but not necessarily 1 (i.e., the path can contain sites in columns that have been drilled). Finally, we will also need to consider the event $\mathcal {A}_z(n)$ that none of the sites in the seed have been drilled in the z-direction.

Note that $\mathcal {S}_B(n)$ only depends on the Bernoulli percolation procedure while $\mathcal {A}_z$ only depends on the columnar mechanism of removal. Therefore they are independent events. Furthermore the occurrence $\mathcal {S}_{B,z}(n)$ is assured by the simultaneous occurrence of $\mathcal {S}_B(n)$ and $\mathcal {A}_z(n)$, therefore, for all $n \ge 1$,

$$\begin{aligned} \mathbb {P}_{p_B,p_z} (\mathcal {S}_{B,z}(n))&\ge \mathbb {P}_{p_z} (\mathcal {A}_z(n))\mathbb {P}_{p_B} (\mathcal {S}_B(n)) \nonumber \\&= p_z^{c^2 \log (n)} \mathbb {P}_{p_B}(\mathcal {S}_B(n)). \end{aligned}$$

(13)

Assume now that there exist $c>0$ and $\delta >0$ such that

$$\begin{aligned} \mathbb {P}_{p_B} (\mathcal {S}_B(n)) \ge \delta , \end{aligned}$$

(14)

for all $n \ge 1$.

Plugging into (13) we get:

$$\begin{aligned} \mathbb {P}_{p_B, p_z} (\mathcal {S}_{B,z}(n)) \ge p_z^{c^2 \log (n)} \delta = \delta n^{-\alpha }. \end{aligned}$$

(15)

for all $n \ge 1$. For the special case $n = L/c$ we obtain:

$$\begin{aligned} \mathbb {P}_{p_b,p_z} \left( \begin{array}{l} \text { there is an open path } \gamma \\ \text {spanning the lattice vertically} \end{array} \right) \ge ({\delta }{c^{\alpha }}) L^{-\alpha } \end{aligned}$$

which is exactly Eq. (12).

From the discussion above, in order to conclude the proof, it is sufficient to find $c>0$ and $\delta >0$ for which (14) holds for all $n \ge 1$. Before we tackle this problem, let us mention that the exponent $\alpha$ above depends on both c and $p_z$. Since the value of c to be fixed latter will depend on $p_B$, we conclude that $\alpha$ actually depends on both $p_z$ and $p_B$.

Let us now move to the proof of (14). The main technique we use is the so-called one-step renormalization or block argument. Roughly speaking it consists of tiling the lattice with cubes of side length c (called blocks) and then working on a new renormalized lattice where the role of the sites are played by the blocks and where two blocks are considered adjacent (neighbors) whenever they share a face.

Note that this new renormalized lattice is isomorphic to the $\mathbb {Z}^3$ lattice: each block in the bulk has six neighbors corresponding to each of its faces shared with other blocks (above, below, in front, behind, on the left and on the right).

Let us denote by S(c, n) the slab shaped region consisting of all the sites in the lattice located above the fixed seed and whose height range from $z=0$ to $z=cn$. Notice that the corresponding region in the renormalized lattice is just an $n \times \log (n)$ rectangle, thus it is strictly two-dimensional. (That’s the reason why we have picked the strip-shaped seed. Other choices would have given a more complicated region.)

For a particular block B in S(c, n), typically, there are 8 other blocks in S(c, n) sharing a face or a line segment with B. Define $\tilde{B}$ as being the union of B and these 8 blocks:

$$\begin{aligned} \tilde{B} = \bigcup _{j,k \in \{0,\pm 1\}} B+jc e_2 + kc e_3, \end{aligned}$$

where $e_2$ and $e_3$ stand for the unit vectores in the z and y direction. (In the case that B does not lie in the bulk of S(c, n) there will be less neighbors, however, the arguments we present go along the same lines.)

For a fixed block B we say that the event $\mathcal {U}(B)$ occurs if the Bernoulli site percolation process restricted to $\tilde{B}$ satisfies:

• 1. There exists a unique cluster $C(B) \subset \tilde{B}$ with diameter greater or equal to c.

• 2. The cluster C(B) intersects every cube of side length c contained in $\tilde{B}$.

• 3. The cluster C(B) touches all the faces of $\tilde{B}$.

Definition 1

When the event $\mathcal {U}(B)$ occurs we say that B is a well-connected block.

Since S(c, n) corresponds in the renormalized lattice to a $\log (n) \times n$ rectangle, one can think of the configuration of well-connected blocks as the realization of a 2-d percolation model in this renormalized lattice. This is not an independent percolation as the state of each block depends on the state of its immediate neighboring blocks. However the dependencies are only finite range. Indeed the events $\mathcal {U}(B_1)$ and $\mathcal {U}(B_2)$ are independent as soon as $\text {dist}(B_1,B_2) >c$.

We now claim that

$$\begin{aligned} \begin{array}{l} \text {If} B_1 \text {and } B_2 \text { are two neighboring} \\ \text {blocks, then}\, \mathcal {C}(B_1) \cap \mathcal {C}(B_2) \ne \emptyset . \end{array} \end{aligned}$$

(16)

In fact, since ${C}(B_1)$ has to touch every cube of side length c contained in $\tilde{B}_1$, ${C}(B_1) \cap \tilde{B}_2$ has diameter greater than c. By the uniqueness of ${C}(B_2)$ inside $\tilde{B}_2$ it has to contain ${C}(B_1) \cap \tilde{B}_2$.

The above claim provides a handy way of gluing the clusters of neighboring well-connected blocks. Indeed, if we find a sequence of neighboring well-connected blocks, (16) then the clusters inside each one of them are part of a larger cluster that extends inside the union of all of them. Since each of the clusters touch the faces of the corresponding block, we can navigate inside this sequence of blocks passing through a path of open sites. We state this as a proposition:

Proposition 1

If there exists a path $B_1, B_2, \ldots , B_k$ of successive neighboring well-connected blocks spanning the renormalized region S(c, n) from top to bottom, then there existis a path $\gamma$ of open sites such that

$$\begin{aligned} \gamma \subset [\mathcal {C}(B_1) \cup \mathcal {C}(B_2) \cdots \cup \mathcal {C}(B_k)] \cap S(c,n) \end{aligned}$$

and that spans S(c, n) from top to bottom. In particular, the event $\mathcal {S}_B(n)$ occurs.

The idea now is to tune c in order to show that a sequence of well-connect blocks can be found with good probability inside the slab S(c, n). For that we first recall that S(c, n) corresponds in the renormalized lattice to a $n\times \log (n)$ rectangle where the process of well-connected blocks presents finite range dependencies. Second, we state the following straightforward fact: for 2-d Bernoulli percolation, the probability of spanning the rectangle $[0, \log {n}] \times [0,n]$ converges to 1 as $n \rightarrow \infty$ provided that the retention parameter p is large enough, say $p \ge p_o$ for some $p_o > p_c^{(2)}$ large enough [35, Theorem 11.55]. The same remains true when Bernoulli percolation is replaced by a given finite-range dependent percolation [36, Theorem 0.0] (may be increasing the value of $p_o$ accordingly).

Proposition 2

There exists a $\delta >0$ and $p_\circ \in (p_c^{(2)}, 1)$ such that, if

$$\begin{aligned} \mathbb {P}_{p_B} \left( \begin{array}{l} \textit{a block }B \textit{ is well-connected } \end{array} \right) \ge p_\circ \end{aligned}$$

then

$$\begin{aligned} \mathbb {P}_{p_B} \left( \begin{array}{l} \textit{there exists a path of well-connected}\\ \textit{blocks spanning } S(c,n) \textit{ vertically} \end{array} \right) > \delta . \end{aligned}$$

(17)

It is now clear, from the Propositions 1 and 2 above, that all we need to do in order to obtain Eq. (14) is to prove that if we choose c very large, then a supercritical percolation process restricted to a cube B of side-length c will fulfil the three items in the definition of the event $\mathcal {U}(B)$ with very high probability (eventually larger than $p_\circ$).

This follows from the fact that, when c is very large then, with high probability there will be a unique cluster occupying a non-negligible fraction of the volume of the cube. Also, with high probability, this cluster will be well spread into the cube and touch all of its faces. This is also a straightforward fact in supercritical Bernoulli percolation that we summarize this in the next:

Proposition 3

There exists $c>0$ such that,

$$\begin{aligned} \mathbb {P}_{p_B} \left( \begin{array}{l} \textit{a block }B \textit{ of side-length}\,c\\ \textit{is well-connected} \end{array} \right) \ge p_\circ . \end{aligned}$$

(18)

We refer the reader to either [37, Theorem 3.1] or [38, Theorem 5] for a rigorous proof of this proposition. There the authors obtain quantitative estimates showing that the probability of $\mathcal {U}(c)$ converges to 1 as c converges to infinity.

Putting together the three previous propositions readily implies that Eq. (14) holds for all n provided that the value of c is chosen sufficiently large.