Results of the RG Technique

In this section we evaluate the utility of the RG methods in estimating the diffusivity. We will make a quantitative comparison of the RG estimates and Monte Carlo estimates ${D}_{\text {MC}}$ on the Bethe lattice below. We will refer frequently to the relative error of $D_{\text {RG}}$ with respect to ${D}_{\text {MC}}$, by which we mean $(D_{\text{RG}}-{D}_{\text{MC}})/{D}_{\text{MC}}$. (For example, see Figs. 4.4, 4.5, and 4.6.) But first we explain the results in a qualitative way by examining the fixed points. We compare our RG to the flow that would result if we were to construct our map with perfect estimates of $D$. Such a flow would follow contours of a plot of the true value of $D$ as a function of $p$ and $\nu$, which we approximate with Monte Carlo estimates in Fig. 4.3.

Figure: Contour plot of Monte Carlo estimates ${D}_{\text {MC}}$ for the Bethe lattice with $z=3$. The irregularities in the lines are artifacts from discrete sampling of parameter space. The value of $D$ for a contour is equal to the value of $p$ at which the line crosses the line $\nu =0$. The effect of the critical point $(p,\nu )=(1/2,1)$ is visible. Some of the curves shown touch the line $\nu =1$ at values of $p$ with $p>p_c$. All points on the line $\nu =1$ with $p<p_c$ are on the contour $D=0$.

cm

Our previous discussion suggests that the fixed points of an RG flow correspond to parameters for which the macroscopic view of the model is invariant under a change of scale. (See for instance [9].) For instance, when applying RG methods to the static percolation problem, one typically finds a one-dimensional map with three fixed points; $p=0,1,$ and $p^*$. (See [26].) In this case, it is the correlation length $\xi$ that we want to hold invariant under iteration of the map. At the points $p=0$ and $p=1$, $\xi$ is 0. The point $p^*$ is an approximation of the critical point, for which $\xi=\infty$. Only at these three points is $\xi$ invariant under a change of length scale. The fixed points of the RG for the FBRW occur for parameter values for which the true behavior of the system is invariant under a rescaling of time and space. At the point $(p,\nu)=(1,1)$, the temporal correlation is infinite and the correlation length $\xi=0$. (If we include the infinite cluster in the computation of the correlation length, then $\xi$ is infinite.) At this point both temporal and spatial correlations are invariant under a change of scale. The point $(p,\nu)=(0,1)$ is invariant for the same reason. When $\tau>0$, we expect spatial correlations to persist over short enough time scales. But for points $(p,0)$ with $0<p<1$, the fluctuations are infinitely rapid so that effects of spatial correlations are destroyed on any finite time scale. Thus, these points are invariant in the actual system as well. However, the maps RGA and RGB are deficient in that there is no fixed point representing the critical point $(p_c,1)$, except in the case $d=1$. This feature limits the utility of these maps in the vicinity of the critical point when $d\ne 1$. When $d=1$, the fixed points $(1,1)$ and $(p_c,1)$ coincide. Furthermore, we will see below that RGA for $d=1$ also predicts the asymptotic behavior in $\tau$ correctly. Thus, the 1-$d$ map captures the important features of the true flow and so produces excellent numerical predictions. RGA$_{m,n}$ does well in this case even for relatively small $m$ and $n$. For instance, it was the success of RGA$_{1,4}$ in the work by LNS [19] that prompted the present study.

This leaves us with the question of how the absence of a critical point in the RG maps affects the predictions of the approximations $D_{\text {RG}}$ when $d>1$ and on the Bethe Lattice. As we mentioned before, the approximations $D^A$ and $D^I$, are quite accurate as long as $\nu$ is not near $1$. So we expect the RG maps to do well in this region. But for $\nu$ near $1$, we expect the absence of a critical point to be manifest. It turns out that the maps in higher dimensions behave qualitatively as they do in one dimension. That is, the flow as $\nu\to 1$ looks roughly as it should in the sub-critical phase. In particular, we will see below that $D_{\text {RG}}$ decays algebraically in $\tau$ to zero for all values of $p$. In the actual model, $D$ decays to a non-zero constant for $p>p_c$. Furthermore, we will see that $D_{\text{RGA}}$ decays as $\tau^{-\alpha }$, where $\alpha$ approaches $1$ as $m$ and $n$ increase. Thus, unless $\tau$ is very large, RGA gives a decay that is close to the true behavior of the model when $p<p_c$, as seen in Figs. 4.5 and 4.6 . This decay, together with an accurate flow away from $\nu =1$ results in good numerical predictions $D_{\text{RGA}}$ when $\nu$ is near $1$ and $p<p_c$. Seen this way, the point $(p,\nu)=(1,1)$ can indeed be viewed as representing the critical point $(p_c,1)$. It is numerically inaccurate, but the flow away from this point represents qualitatively (and even somewhat numerically) the flow away from $(p_c,1)$ in the actual system. Because RGB is based on much more accurate estimates of $D$ than is RGA, we will find that RGB does much better when $\nu$ is not close to $1$. However, RGB$_{m,n}$ for large $m$ and $n$ predicts a decay in $\tau$ with an exponent much larger than $1$, so it makes poor predictions when $\nu$ is close to $1$. These observations are illustrated in Figs. 4.4 and 4.5 .

Figure: Relative error of $D_{\text {RG}}$ compared to ${D}_{\text {MC}}$ for the Bethe lattice. Here, $E=\log_{10}(\vert D_{\text{RG}}-{D}_{\text{MC}}\vert/{D}_{\text{MC}})$. The curves are computed for several values of $p$, using RGA$_{9,10}$ and RGB$_{9,10}$. All maps do best when $\tau$ is small, but RGB is much better than RGA here. For the RGA shown here $\alpha =$, so that for $p$ well below $p_c$ the estimate of $D$ is good even for large values of $\tau$. For $p>p_c$ (eg. $p=0.9$), the RG maps predict an algebraic decay to zero, while the Monte Carlo result predicts decay to a constant. Notice the statistical fluctuations in ${D}_{\text {MC}}$, which are of order $10^{-5}$-$10^{-6}$ times the value of ${D}_{\text {MC}}$.

cm

Figure: Comparison of Monte Carlo estimates ${D}_{\text {MC}}$ to RG estimates $D_{\text {RG}}$ for RGA$_{9,10}$ and RGB$_{9,10}$ on the Bethe lattice, for which $p_c=0.5$. Squares are ${D}_{\text {MC}}$, pluses are for RGA$_{9,10}$, and diamonds are for RGB$_{9,10}$. For $p<p_c$, the diffusivity ${D}_{\text {MC}}$ decays to 0. For $p>p_c$, it tends to a constant. The RG estimates decay to 0 both above and below $p_c$. As $p$ increases, the onset of the algebraic decay in the RG estimates begins for larger values of $\tau$.

cm

Figure: Comparison of Monte Carlo estimates ${D}_{\text {MC}}$ to RG estimates $D_{\text {RG}}$ for RGA$_{9,10}$ and RGB$_{9,10}$ on the Bethe lattice, for which $p_c=0.5$. Squares are ${D}_{\text {MC}}$, pluses are for RGA$_{9,10}$, and diamonds are for RGB$_{9,10}$. For $p<p_c$, all three estimates show algebraic decay; i.e. $D\propto \tau ^{-\alpha }$. For ${D}_{\text {MC}}$, $\alpha \approx 1$. For the RGs, $\alpha$ is known exactly. For RGA$_{9,10}$, $\alpha \approx 0.9997$. For RGB$_{9,10}$, $\alpha \approx 10$. For small $p$, RGA$_{9,10}$ is close to ${D}_{\text {MC}}$ when they both begin to decay at nearly the same rate, resulting in a good numerical estimate for rather large $\tau$. For $p>p_c$, the RG predictions continue to decay algebraically, while ${D}_{\text {MC}}$ tends to a constant.

cm

Details of the performance of the various RG methods depend in a complicated way on the model parameters and $m$ and $n$ and the dimension of the lattice. Now that we have desribed the broad features of the flow, we will attempt to fill in some details. In general, we get the best results from the RG maps when $m$ and $n$ in (4.2), (4.3), and (4.4) are as large as possible (with $m<n$). Typical values are $m=9$ and $n=10$. The relative error generally decreases both with increasing $m$ and increasing $n$. However, for RGA, the error changes sign as a function of the parameters. So this rule is violated for many values of $\tau$ and $p$. RGB does much better than RGA when $\tau$ is not too large. When $\tau$ is large enough (much greater than $m$ and $n$), RGA does much better than RGB if $p<p_c$ although RGA fails eventually for large enough $\tau$. If $p>p_c$ and $\tau$ increases past $n$, both RGA and RGB begin to fail, with RGB performing slightly better. However, as $p$ becomes larger, both maps fail less rapidly with increasing $\tau$. Both RGA and RGB fail to capture even the qualitative behavior for $p>p_c$ and $\tau$ very large. In particular, RGA and RGB both predict that $D_{\text{RG}}\to 0$ as $\tau$ increases, while ${D}_{\text {MC}}$ and previous studies show that the diffusivity actually decays to a non-zero constant. In general, the relative error of all RG maps increases without bound with increasing $\tau$. We see evidence of this behavior in the Monte Carlo data (Fig. 4.4) as well as in the asymptotic analysis discussed below. (We will also discuss some exceptional cases in which the relative error approaches a limit. Also, there is a brief dip in the error for RGA as the sign of the error changes.) The relative error in RGA decreases with increasing $p$ when $\tau$ is not too large. When $\tau$ is large enough, the error first decreases then begins to increase when $p$ is still less than $p_c$, then begins to decrease again when $p>p_c$. The relative error in RGB decreases as $p$ moves away from the critical point in either direction and toward 0 and $1$. Except when $p$ is very near $1$, RGB$_{m,n}$ has a much lower relative error than RGA$_{m,n}$ for $\tau$ small enough. If $m$ is not much smaller than $n$, then this regime includes values of $\tau \lessapprox n$.

We need to ask whether the RG maps give estimates that improve on the approximations $D^A_n$ and $D^I_n$, that is, the approximations on which the maps are based. The short answer is yes, when $\tau$ is not too large, as illustrated in Fig. 4.7. It is appropriate to compare the relative error in RGA$_{9,10}$ to that in $D^A_{10}$, and the relative error in RGB$_{9,10}$ to that in $D^I_{10}$. (One could argue that we should use $D^A_{9}$ and $D^I_{9}$, but the results are quite similar.) We see that, for $p<p_c$, RGA$_{m,n}$ improves on $D^A_n$ for all $\tau$. For values of $\tau$ that are larger than those plotted, we expect this behavior to continue, because RGA$_{m,n}$ decays algebraically in $\tau$, as does the true value of $D$, while $D^A_n$ approaches a non-zero limit. For $p>p_c$ RGA is not an improvement for large $\tau$. We expect this because the true value of $D$ decays to a non-zero limit in this case. Comparing RGB$_{m,n}$ and $D^I_n$ we see much the same behavior, except that RGB$_{m,n}$ does not give as great an improvement as RGA$_{m,n}$ does for large $\tau$ and $p<p_c$. Also, when $\tau$ is not too large, both RGB$_{m,n}$ and $D^I_n$ have much smaller relative error than do RGA$_{m,n}$ and $D^A_n$, as is expected. The general features of the performance of the RG maps relative to the approximate diffusivities, which are discussed above, do not depend $m$ and $n$. Similar curves are also obtained for different values of $p$, with the same qualitative behavior depending on whether $p$ is greater than or less than $p_c$. Particularly large and small values of $p$ were chosen for Fig. 4.7 in order that the curves be well separated.

Figure: Comparison of relative error in $D_{\text {RG}}$ to relative error in $D^A$ and $D^I$ on the Bethe lattice for various values of $p$. For all curves, $E=\log_{10}(\vert D_x-{D}_{\text{MC}}\vert/{D}_{\text{MC}})$, where $D_x$ is one of RGA$_{9,10}$, RGB$_{9,10}$, $D^A_{10}$ or $D^I_{10}$. Upper plot: curves marked $D$ are for $D^A_{10}$; curves marked rga are for RGA$_{9,10}$. Lower plot: curves marked $D$ are for $D^I_{10}$; curves marked rgb are for RGB$_{9,10}$. When $\tau$ is less than about $10$ or $100$, the RG maps improve on the estimates $D^A$ and $D^I$. The RG maps, generally perform worse for larger $\tau$. An exception is RGA when $p<p_c$, in which case RGA predicts algebraic decay in $\tau$ with nearly the correct exponent, while the estimate $D^A$ incorrectly decays rapidly to a constant. Notice that the statistical noise in the Monte Carlo data when both $\tau$ and the error are small obscures the relative position of the curves.

cm

cm cm