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# 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 on the Bethe lattice below. We will refer frequently to the relative error of with respect to , by which we mean . (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 . Such a flow would follow contours of a plot of the true value of as a function of and , which we approximate with Monte Carlo estimates in Fig. 4.3.

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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; and . (See [26].) In this case, it is the correlation length that we want to hold invariant under iteration of the map. At the points and , is 0. The point is an approximation of the critical point, for which . Only at these three points is 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 , the temporal correlation is infinite and the correlation length . (If we include the infinite cluster in the computation of the correlation length, then is infinite.) At this point both temporal and spatial correlations are invariant under a change of scale. The point is invariant for the same reason. When , we expect spatial correlations to persist over short enough time scales. But for points with , 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 , except in the case . This feature limits the utility of these maps in the vicinity of the critical point when . When , the fixed points and coincide. Furthermore, we will see below that RGA for also predicts the asymptotic behavior in correctly. Thus, the 1- map captures the important features of the true flow and so produces excellent numerical predictions. RGA does well in this case even for relatively small and . For instance, it was the success of RGA 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 when and on the Bethe Lattice. As we mentioned before, the approximations and , are quite accurate as long as is not near . So we expect the RG maps to do well in this region. But for near , 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 looks roughly as it should in the sub-critical phase. In particular, we will see below that decays algebraically in to zero for all values of . In the actual model, decays to a non-zero constant for . Furthermore, we will see that decays as , where approaches as and increase. Thus, unless is very large, RGA gives a decay that is close to the true behavior of the model when , as seen in Figs. 4.5 and 4.6 . This decay, together with an accurate flow away from results in good numerical predictions when is near and . Seen this way, the point can indeed be viewed as representing the critical point . It is numerically inaccurate, but the flow away from this point represents qualitatively (and even somewhat numerically) the flow away from in the actual system. Because RGB is based on much more accurate estimates of than is RGA, we will find that RGB does much better when is not close to . However, RGB for large and predicts a decay in with an exponent much larger than , so it makes poor predictions when is close to . These observations are illustrated in Figs. 4.4 and 4.5 .

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Details of the performance of the various RG methods depend in a complicated way on the model parameters and and 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 and in (4.2), (4.3), and (4.4) are as large as possible (with ). Typical values are and . The relative error generally decreases both with increasing and increasing . However, for RGA, the error changes sign as a function of the parameters. So this rule is violated for many values of and . RGB does much better than RGA when is not too large. When is large enough (much greater than and ), RGA does much better than RGB if although RGA fails eventually for large enough . If and increases past , both RGA and RGB begin to fail, with RGB performing slightly better. However, as becomes larger, both maps fail less rapidly with increasing . Both RGA and RGB fail to capture even the qualitative behavior for and very large. In particular, RGA and RGB both predict that as increases, while 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 . 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 when is not too large. When is large enough, the error first decreases then begins to increase when is still less than , then begins to decrease again when . The relative error in RGB decreases as moves away from the critical point in either direction and toward 0 and . Except when is very near , RGB has a much lower relative error than RGA for small enough. If is not much smaller than , then this regime includes values of .

We need to ask whether the RG maps give estimates that improve on the approximations and , that is, the approximations on which the maps are based. The short answer is yes, when is not too large, as illustrated in Fig. 4.7. It is appropriate to compare the relative error in RGA to that in , and the relative error in RGB to that in . (One could argue that we should use and , but the results are quite similar.) We see that, for , RGA improves on for all . For values of that are larger than those plotted, we expect this behavior to continue, because RGA decays algebraically in , as does the true value of , while approaches a non-zero limit. For RGA is not an improvement for large . We expect this because the true value of decays to a non-zero limit in this case. Comparing RGB and we see much the same behavior, except that RGB does not give as great an improvement as RGA does for large and . Also, when is not too large, both RGB and have much smaller relative error than do RGA and , 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 and . Similar curves are also obtained for different values of , with the same qualitative behavior depending on whether is greater than or less than . Particularly large and small values of were chosen for Fig. 4.7 in order that the curves be well separated.

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Next: Behavior of the RG Up: The Renormalization Group Technique Previous: The RG technique   Contents
John Lapeyre 2003-12-09