Growth by random walker sampling and scaling of the dielectric breakdown model

Random walkers absorbing on a boundary sample the harmonic measure linearly and independently: we discuss how the recurrence times between impacts enable nonlinear moments of the measure to be estimated. From this we derive a technique to simulate dielectric breakdown model growth, which is governed nonlinearly by the harmonic measure. For diffusion-limited aggregation, recurrence times are shown to be accurate and effective in probing the multifractal growth measure in its active region. For the dielectric breakdown model our technique grows large clusters efficiently and we are led to significantly revise earlier exponent estimates. Previous results by two conformal mapping techniques were less converged than expected, and in particular a recent theoretical suggestion of superuniversality is firmly refuted.

Figure 1

The moment spectrum $\tau \left(q\right)$ of the harmonic measure obtained over the growth of a single DLA cluster by imposing the condition $Z(q,t,N)-Z(q,t,N∕4)=Z(q,t,N∕4)-Z(q,t,N∕16)$ to yield $t=\tau \left(q\right)∕D$. The respective curves used ages of order $k=1,2,4,8$ and can be seen to break away around $q=k+1$ as expected from our discussion in Sec. 2. Note also the close agreement with the electrostatic scaling law [23] that $\tau \left(3\right)∕D=1$. These results come from a single off-lattice DLA cluster of $N={10}^6$ particles grown with noise reduction factor [15] $A=0.1$.

Figure 2

Multifractal spectrum of (the harmonic measure of) DLA obtained using ages of order $k=1,3,9,27$. Also shown are earlier results from Ball and Spivack [24] and Jensen et al. [25]. Note how much better the present data agree with the tangent lines representing Halsey’s electrostatic scaling law (for $q=3$) and Makarov’s theorem (for $q=1$). The inset shows how the older data extend to higher $\alpha$, which our method cannot probe.

Figure 3

Size dependence of the $f\left(\alpha \right)$ spectrum of DLA, obtained using first-order ages and clusters grown per Fig. 2 to different sizes. In the vicinity $\alpha \lesssim D$ the curves systematically rise with the range of $N$ used, in support of our claim that the age method ultimately probes all the way to $\alpha =D$. The curves are based on the slope of $\ln Z(q,0,N)$ vs $\ln N$ across successive factors of $\sqrt{10}$ in $N$ up to $N={10}^7$ (topmost curve).

Figure 4

Scatter plot of the ages of sites hit for DLA: $\eta =1$, $\delta Q=1$. The data points are decimated for large $N$ for clarity. The solid lines correspond to the expected scaling of minimal ages obtained from Eqs. (4a, 4b); these theoretical estimates appear to be suitably cautious.

Figure 5

Study of the highest probability growth sites and the tips of DLA clusters. We launched probe walkers onto a fixed nongrowing cluster and recorded the number of probes landed on each site. This enables us to calculate the growth probability of each site, including the tip (farthest site from the origin). (a) The ratio of the growth probabilities of the highest hit rate site and the tip: it is a slowly increasing function of the cluster mass $M$. The best fit curve is a power law with a small exponent: $p_{\max }∕p_{\text{tip}}\sim M^{({\alpha }_{\text{tip}}-{\alpha }_{\min })∕D}$, with ${\alpha }_{\text{tip}}-{\alpha }_{\min }=0.03\pm 0.005$. However, one cannot entirely exclude logarithmic corrections (see inset); in that case ${\alpha }_{\text{tip}}={\alpha }_{\min }$. (b) The number of sites with higher hit rate than the tip. This scales as a power of the radius $R$ with the fractal dimension of tips as exponent: $n_{p⩾p_{\text{tip}}}\sim R^{f_{\text{tip}}}\sim M^{f_{\text{tip}}∕D}$. The exponent is larger, clearly distinct from zero: $f_{\text{tip}}=0.38\pm 0.03$. On all plots the two data sets are standard DLA $(A=1)$ with ${10}^4–{10}^5$ clusters depending on size and noise-reduced DLA $(A=0.1)$ with $4200–{10}^4$ clusters. The number of probes was set such that the tip of each cluster received 2500 hits. The solid lines are best fit for the data points under them; dotted lines are extensions towards excluded data points.

Figure 6

DBM clusters grown by the new method: (a) $\eta =0.5$, (b) $\eta =1.5$, (c) $\eta =2$, and (d) $\eta =3$. In each case $N={10}^6$ random walkers were used.

Figure 7

(Color online) Comparison of the relative penetration depth $\Xi =\sqrt{⟨r^2⟩-{⟨r⟩}^2}∕⟨r⟩$ for the walker-DBM and HL methods. For each method the penetration depth was measured on 30 DBM clusters, $\eta =2$, with 3000 probes on each cluster. Both methods converge to the same asymptotic value, with the walker-DBM converging slightly faster.

Figure 8

(Color online) (a) The tip scaling exponent ${\alpha }_{\text{tip}}$ and (b) the fractal dimension $D$ as a function of $\eta$. The four data sets plotted are the walker-DBM, our HL measurements, earlier HL measurements of Ref. 17, and our Shraiman-Bensimon results. Because we have to convert the SB results back from measurements at fixed ${\eta }_1$ to the corresponding values of ${\eta }_0={\eta }_1∕\alpha$, these points have error bars on both $\alpha$ and ${\eta }_0$. The “walker” data point for $\eta =1$ is in fact a DLA measurement (with very small uncertainties) from Ref. 27 together with the relation ${\alpha }_{\text{tip}}=D-1$. The dotted lines correspond to the limiting behaviors: dense two-dimensional growth $(\eta =0)$ and nonbranching one-dimensional growth $(\eta ⩾{\eta }_c)$.

Figure 9

(Color online) Direct integration of the Bensimon-Shraiman equation (6) as described in Sec. 5 leads to the prediction that $v_{\mathrm{cum}}\left(k\right)={\sum }_{j<k}{\mid {\psi }_j\mid }^2$ should vary as $k^{{\eta }_1(1-1∕\alpha )}$ so the displayed graphs should exhibit slope $1-1∕\alpha$. (Recall that ${\eta }_1=\alpha \eta$.) The upper panel (a) shows plots from the same simulation at ${\eta }_1=0.5$ for successively later-times (bottom to top). There is characteristic slope at high $k$ which persists over a limited range, but it is the lower slope which emerges and eventually dominates the wider range down to the smallest range of $k$ and which we take to reveal the true tip scaling exponent $\alpha$. The lower panel (b) shows data at large times for ${\eta }_1=0.3,0.6,0.9,1.2,1.5,1.8$ (top to bottom at right). While the high-$k$ slope is insensitive to ${\eta }_1$, there is a clear variation of the asymptotic slopes from which we take $\alpha$ in Fig. 8a.