Random Walks on the Static Lattice

In this section, we discuss random walks on the static lattice (i.e. on a percolation process), giving particular attention to the critical behavior. A review of the subject can be found in reference [25]. We consider a random walk on the bond percolation process on ${\Bbb Z}^d$. The process is equivalent to the fluctuating bond random walk when the bond fluctuation time $\tau=\infty$. One can imagine executing a random walk on a realization of the percolation process on an infinite lattice. However it is useful to review the process in the context of the FBRW. The walker begins at the origin and attempts to step at unit time intervals from the site it occupies to one of its nearest neighbors chosen at random. If the bond connecting the occupied site to the chosen nearest neighbor has not been attempted before, then this bond's state is chosen at random; it is open with probability $p$ and closed with probability $1-p$. If bond is determined to be closed then the walker stays at the occupied site. But if the bond is open, the walker crosses it and occupies the chosen nearest neighbor. Once the state of a bond has been chosen, it remains in that state for all time.

As expected, the critical phenomenon in the percolation process determines different qualitative regimes for the random walk. The walk is, in effect, a walk on the open cluster at the origin. For $p<p_c$, the cluster at the origin is finite with probability one, and the cluster size distribution decays rapidly (equation 2.7 ), so that the mean square displacement of the walker cannot increase without bound. For $p=p_c$, there is no infinite cluster at the origin, but almost; the distribution of finite clusters has an algebraically decaying tail. Furthermore, as discussed above, there is no scale on which a large cluster is homogeneous. As the walker wanders farther from the origin, it encounters ever larger blobs to get lost on and ever larger holes to go around. One can think of the lattice as having different densities on different length scales, and thus a diffusivity that varies with the length scale. Thus asymptotically, the mean square displacement is not proportional to the time $n$, but is proportional to $n$ to some power less than $1$. Or, in the case of the Bethe lattice, $\langle S^2 \rangle$ grows logarithmically. A simulation on the Bethe lattice for $z=3$ is shown in Fig. 2.4.

Figure: Mean square displacement on the static lattice at the percolation threshold $p=p_c$. The data is from Monte Carlo simulations on the Bethe Lattice with $z=3$. Although, it is well known that no infinite cluster exists at $p=p_c=1/2$, the figure shows that the mean square displacement increases logarithmically with time. The average is over $10^5$ trials. The metric used is given by (2.44).

When $p>p_c$, there are two possibilities for a realization of the walk; the walker begins either on the infinite cluster or on a finite cluster. The distribution of the largest clusters decays rapidly so that realizations of the walk on these clusters do not contribute to the asymptotic mean square displacement. Only the paths on the infinite cluster contribute. When the walker has traveled a distance much farther than $\xi$, the infinite lattice appears homogeneous, so we expect normal diffusion, but with a reduced diffusivity. Below, we consider each of these cases and note the relevant critical exponents.

When $p<p_c$ and the bonds are static ($\nu =1$), the mean square displacement approaches a limit as the time $n\to\infty$. The interesting quantity in this case is the asymptotic mean square displacement. When $n=\infty$ the walker occupies each site on the cluster at the origin $C$ with equal probability. So the limiting mean square displacement on $C$ is a kind of mean square radius

$$R^2(C) = \sum_{x\in C} \frac{\Vert x\Vert^2}{\vert C\vert},$$ (2.14)

where $\Vert x\Vert$ is the distance of the site $x$ from the origin. Note that this radius depends on the position of the cluster relative to the origin. The mean square displacement is then $R^2(C)$ averaged over all clusters

$$\langle S^2_\infty \rangle = \left<{R^2(C)}\right> = \sum_c P(C=c) R^2(c),$$ (2.15)

where $c$ is a particular realization of the cluster at the origin, and the sum is over all possible clusters. Equation 2.15 gives the asymptotic mean square displacement of the random walk on a sub-critical percolation process. We write $\langle S^2_\infty \rangle$ in a form convenient for calculations by defining a mean square radius averaged over all clusters of size $\vert C\vert=s$,

$$R_s^2 = \frac{\sum_{\{c:\vert c\vert=s\}} P(C=c) R^2(c)}{P(\vert C\vert=s)},$$ (2.16)

so that (2.15) becomes

$$\langle S^2_\infty \rangle = \sum_s R^2_s \ P(\vert C\vert=s).$$ (2.17)

Now we assume that, for the range of $s$ that we are interested in, $R^2_s$ obeys a single power law

$$R^2_s \approx s^{2\epsilon}.$$ (2.18)

In the sums we consider, such as (2.17), we will see that the main contribution comes from large $s$, but $s$ smaller than the crossover mass $(p_c-p)^{1/\sigma}$. Thus $s$ is in the regime where the clusters have a fractal structure. So we expect $\epsilon\ne 1/d$, even if the dominant clusters are roughly spherical.

We find the critical exponent of $\langle S^2_\infty \rangle$, using the scaling ansatz for the cluster mass (2.11). Furthermore, we express the exponent $\epsilon$ in terms of the other critical exponents. To do this, we use $R^2_s$ in an expression for the correlation length that involves (2.11). Thus, we can express $\epsilon$ as a function of $\sigma$, $\theta$, and $\phi$. Under reasonable assumptions the definition of correlation length (2.9) is equivalent to

$$\xi^2 = \frac{\sum_{x\in{\Bbb Z}^d} \Vert x\Vert^2 P(x\in C)} {\sum_{x\in{\Bbb Z}^d} P(x\in C)}.$$ (2.19)

In order to rewrite this equation in terms of cluster mass, we use the indicator function. Letting $\omega$ be a particular realization of the percolation process, and letting $A$ be a (measurable) event, we define

$$1_A(\omega) = \begin{cases}1& \text{if event $A$\ occurs for configuration $\omega$}, \\ 0 & \text{otherwise} \ . \end{cases}$$ (2.20)

It is easy to see that $\left<{1_A}\right>=P(A)$. Then we have

$$\sum_{x\in{\Bbb Z}^d} \Vert x\Vert^2 P(x\in C) = \sum_{x\in{\Bbb Z}^d} \Vert x\Vert^2 \left<{1_{x\in C}}\right>$$ (2.21)

$$= \left<{\sum_{x\in{\Bbb Z}^d} \Vert x\Vert^2 1_{x\in C}}\right>$$ (2.22)

$$= \left<{\sum_{x\in C} \Vert x\Vert^2}\right>$$ (2.23)

$$= \sum_s s P(\vert C\vert=s) \frac{\sum_{\{c:\vert c\vert=s\}} P(C=c) \sum_{x\in C} \frac{\Vert x\Vert^2}{\vert C\vert} }{P(\vert C\vert=s)}$$ (2.24)

$$= \sum_s s P(\vert C\vert=s) R^2_s \ .$$ (2.25)

In a similar manner, we have that

$$\sum_{x\in{\Bbb Z}^d} P(x\in C)=\sum_s s P(\vert C\vert=s)=\left<{\vert C\vert}\right>.$$ (2.26)

So we rewrite (2.19) as

$$\xi^2 = \frac{\sum_s s P(\vert C\vert=s) R^2_s } {\sum_s s P(\vert C\vert=s)}.$$ (2.27)

Because we assume that $R^2_s \approx s^{2\epsilon}$, both the numerator and the denominator are moments of $P(\vert C\vert=s)$. We use the scaling ansatz (2.11) to compute the $k$th moment of $P(\vert C\vert=s)$ as

$$\begin{aligned}\sum_s s^k P(\vert C\vert=s) &= \sum_s s^{k-\t... ...fty dz \ z^\frac{+k-\theta+1-\sigma}{\sigma}f(z), \end{aligned}$$

where $z=(p_c-p)s^\sigma$ if $p<p_c$ and $z=(p-p_c)s^\sigma$ if $p>p_c$. Here, we assume that $k-\theta\ge-1$ so that the sum diverges until the rapid decay of $f(z)$ begins. Thus, as $p$ approaches $p_c$ the major contribution of the sum occurs for large $s$ and we are justified in replacing the sum with an integral. Then the critical exponent of the $k$th moment of $P(\vert C\vert=s)$ is $(-k+\theta-1+\sigma)/\sigma.$ Now we evaluate the critical exponent in (2.27). The numerator has $k=2\epsilon+1$, while the denominator has $k=1$. So $\xi^2$ diverges with the power $-2\epsilon/\sigma$. Because we defined $\phi$ via $\xi\approx \vert p-p_c\vert^{-\phi }$, we have that $-2\phi =-2\epsilon/\sigma$, or $\epsilon=\sigma\phi$.

The expression for $\langle S^2_\infty \rangle$ given in (2.17) is the $k$th moment of $P(\vert C\vert=s)$, with $k=2\epsilon$. Thus, the critical exponent for this moment is $-2\phi +(\theta-1+\sigma)/\sigma$. A similar exercise shows that the critical exponent for the strength of the infinite cluster (2.12) is $\beta = (\theta-1+\sigma)/\sigma$. So, for $p<p_c$, we have

$$\langle S^2_\infty \rangle \approx (p_c-p)^{\beta-2\phi }.$$ (2.29)

The asymptotic mean square displacement $\langle S^2_\infty \rangle$ can be calculated exactly in one dimension and on the Bethe lattice. In one dimension one has

$$\langle S^2_\infty \rangle = \frac{p}{(1-p)^2}.$$ (2.30)

Straley [29] has calculated that on the Bethe lattice with $z=3$

$$\langle S^2_\infty \rangle = \frac{3}{2p} \left(3p^2-4p+4(1-p)^2\ln\frac{1-p}{1-2p}\right).$$ (2.31)

We performed Monte Carlo simulations that agree with (2.30) and (2.31).

Now we turn our attention to the walk on the static disordered lattice when $p\ge p_c$. As in the sub-critical case, scaling arguments and numerous Monte Carlo studies provide a rather solid picture of the phenomenology as $p$ approaches $p_c$ from above. When $p=1$ all bonds are open and we have an ordinary random walk with $D=1$. When $p$ is only a bit less than $1$, there are only a few holes of size $\xi$, which is rather small. After a few time steps, when the distribution of the position has spread over a length scale greater than $\xi$, we find that the walk is again diffusive with a slightly reduced diffusivity. In the discussion above we saw that $D=0$ for $p<p_c$. If we believe that there is only one critical point $p_c$, we expect that $D(p)$ is continuous for $p>p_c$. Furthermore, because at $p=p_c$ there is no infinite cluster and the large clusters have a fractal structure, it seems likely that $D(p_c)=0$. Indeed, simulations show that $D(p)$ is continuous for $0\le p\le 1$. In particular, $D(p)$ appears to obey a power law,

$$D(p) \approx (p-p_c)^\mu p\downarrow p_c.$$ (2.32)

The critical exponent $\mu$ is sometimes called a dynamic critical exponent. This is because $\mu$ arises when a dynamic process is added to the percolation process and attempts to derive $\mu$ by considering only percolation have failed. It is widely believed that $\mu$ is also the critical exponent for conductivity on the percolation process. Finally, we ask for the behavior of the walk at the critical point. At the critical point $p=p_c$ it appears that the cluster size distribution decays slowly enough that the expectation of the mean square displacement $\left<{S^2_n}\right>$ is not bounded. However, the walk exhibits anomalous diffusion, that is $\left<{S^2_n}\right>$ grows as $t$ to a power less than one. See, for example, Fig. 2.4.

We relate the behavior in all three regimes via a scaling assumption. The new scaling exponents will be expressed as functions of the static exponents and the dynamic exponent $\mu$. We noted above that, on scales smaller than $\xi$, the percolation process near $p_c$ is indistinguishable from the process at $p_c$. Only on scales greater than $\xi$ can one distinguish, for instance, on which side of the percolation threshold $p$ lies. So if the time $n$ is large, but small enough that the walker has not wandered farther than $\xi$, we should see anomalous diffusion, as on the critical process. For much larger times, we expect to see the diffusive behavior if $p>p_c$, or a bounded mean square displacement if $p<p_c$. It is believed that the correlation function $P(x\in C)$ obeys a scaling assumption similar to (2.11), with the displacement of $x$ as the independent variable. Because time varies as distance to a power, with the power taking different values in different regimes, it is reasonable that the mean square displacement should also satisfy a scaling assumption,

$$\langle S^2_n \rangle ^{1/2} \sim n^k r[(p-p_c) n^x],$$ (2.33)

where $r(z)$ is a scaling function, and $n$ is the time. In the scaling assumptions that we have encountered, a crossover scale separates regions of algebraic and exponential decay. The situation is somewhat different here, in that we require $r(z)$ to behave like a power of $z$ for large $z$. If $z$ is large and positive, the walker is diffusing on the super-critical lattice over distances greater than $\xi$. In this case, $\langle S^2_n \rangle ^{1/2}\propto (p-p_c)^{\mu/2}$, according to (2.32), so we must have $r(z)\propto (p-p_c)^{\mu/2}$. Then $\langle S^2_n \rangle ^{1/2} \approx n^{k+\mu x/2}$. Because the walk is diffusive in this regime, we must have $k+\mu x/2=1/2$, which gives one equation relating $k$ and $x$. Similarly, when $p<p_c$, we must have $r(z)\propto (-z)^{-k/x}$ for $-z\gg 1$ in order that $\langle S^2_n \rangle ^{1/2}$ be independent of the time $n$. Then, in order that the dependence on $p$ agrees with the static limit (2.29), we require $k/x=\phi -\beta/2$. Solving the two equations for $k$ and $x$, we find that

$$x = \frac{1}{2\phi +\mu-\beta},$$ (2.34)

and

$$k=\frac{\phi -\beta/2}{2\phi +\mu-\beta}.$$ (2.35)

At the critical point, we have $\langle S^2_n \rangle ^{1/2}\approx n^k$. The time scale separating the two types of motion is $\vert p-p_c\vert^{-1/x}$. For times that are long, but shorter than this time scale, anomalous diffusion with exponent $k$ is observed. At longer times, the motion either becomes bounded for $p<p_c$, or diffusive for $p>p_c$.