Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model

We present simulations of the one-dimensional Oslo rice pile model in which the critical height at each site is randomly reset after each toppling. We use the fact that the stationary state of this sand-pile model is hyperuniform to reach system of sizes $>{10}^7$. Most previous simulations were seriously flawed by important finite-size corrections. We find that all critical exponents have values consistent with simple rationals: $\nu =\frac{4}{3}$ for the correlation length exponent, $D=\frac{9}{4}$ for the fractal dimension of avalanche clusters, and $z=\frac{10}{7}$ for the dynamical exponent. In addition, we relate the hyperuniformity exponent to the correlation length exponent $\nu$. Finally, we discuss the relationship with the quenched Edwards-Wilkinson model, where we find in particular that the local roughness exponent is ${\alpha }_{\mathrm{loc}}=1$.

Figure 1

A typical avalanche of boundary-driven Oslo model (time flows towards right). (a) The stress profile: $z_i\le 2$ are marked as gray (different shades) and sites with $z_i>2$ are marked as red. (b) A portion of the avalanche, marked as a rectangle in (a), is zoomed to view multiple topplings (marked differently).

Figure 2

Number of topplings on a semi-infinite lattice driven at its left end, until a site at distance $i_{\max }$ from this end is first toppled. Each curve is based on $\approx 1000$ runs. The dashed red and black lines with slopes 2 and 3 are shown for comparison.

Figure 3

Average density in the active region, plotted against an inverse power of the size $i_{\max }$ of the active region, for different initial configurations. Each curve is based on a single run. For all curves, the stationary density is reached as $i_{\max }\to \infty$, but the speed of convergence and the fluctuations depend strongly on the choice of initial state. Periodic initial states with density close to the stationary one are optimal. Notice that what looks like a horizontal straight line is indeed the data for $z_0=272/157=1.732484$.

Figure 4

Average stress $\langle z\rangle$ of driven systems with open boundaries as a function of $L$. Main: log-linear plot of the raw data. Inset: the same data plotted such that one can determine more precisely the finite-size corrections. More precisely, on the $x$ axis is plotted $L^{-\sigma }$ with $\sigma$ either 0.76, 0.74, or 0.72. If $\sigma$ were equal to the exponent of the leading finite-size correction, then the plot of $\langle z\rangle$ versus $L^{-\sigma }$ would be asymptotically a straight line. Since finite-size corrections are very large, data would be indistinguishable from straight lines in such a plot for a wide range of $\sigma$. Therefore, we add to $\langle z\rangle$ a term linear in $L^{-\sigma }$, such that the curves become roughly flat near the origin. The curve which is most straight at the origin gives then the true correction to scaling exponent.

Figure 5

The inset shows the stationary profile $\rho \left(i\right)$ for $L=64$. To verify that it is indeed left-right symmetric, we show in the main plot differences between values of $\rho \left(i\right)$ measured at the same site $i$, but for runs where the system was driven at different values $j_0$ resp. $k_0$. The estimated statistical error for these differences was between $\pm 0.0001$ and $\pm 0.0002$.

Figure 6

Log-log plot of stress depletion near the end points versus distance from the end [more precisely from the point $x=\frac{1}{2}$, halfway between the site $i=0$ where $\rho \left(i\right)$ is zero, and the site $i=1$ where it first becomes positive]. Lattices ($L\ge 2^{15}$) are sufficiently large so that finite-size corrections should be negligible.

Figure 7

The main plot shows a log-log plot of $\mathrm{Var}\left[Z\right]$ versus $L$, together with a straight line that suggests $\mathrm{Var}\left[Z\right]\sim L^{1/2}$ with visible corrections. The inset suggests that these corrections decrease roughly as $L^{-1/2}$.

Figure 8

Avalanche size distributions for the open systems driven at the left boundary. (a) Shows the raw data (for the largest lattices sizes only, to avoid overcrowding). (b) Shows the same data multiplied by $s^{14/9}$. If this exponent is equal to $\tau$, the central parts of the curves should be flat. In view of the substantial corrections to this, we plotted in (c) the data (for smaller $L$, since they have less statistical fluctuations) multiplied by a further factor $(1+a/s^x)$, with $x=0.4$. The two straight lines in (c) indicate the error margins $\pm 0.0005$ of $\tau$. The numbers of avalanches used for these figures range between $>2\times {10}^{10}$ for $L\le 4096,1.7\times {10}^9$ for $L=131072$, and $3.5\times {10}^9$ for $L=262144$. The fluctuations seen for $s<100$ are not statistical, but are systematic, and are related to the structure of the state-space of recurrent stable configurations [36].

Figure 9

Rescaled second moments of avalanche sizes for the boundary model, plotted against $1/L^{0.9}$. The moments are divided by powers of $L$, such that the curve would be straight if the power were equal to $D+1$ and the correction to scaling exponent were $\omega =0.9$. This is indeed the case for $D=\frac{9}{4}$. The other two curves (with exponents $\frac{9}{4}\pm 0.007$) are clearly subcritical and supercritical.

Figure 10

(a) Log-log plot of $P_r(r,L)$, the spatial size distribution of avalanches for boundary-driven systems, multiplied by $r^{9/4}$. (b) The same data, but multiplied by $1+8.3/r^{0.9}$ and plotted on a log-linear plot. The straight lines indicate the estimated errors $\pm 0.002$.

Figure 11

Log-log plot of $P_t(t,L)$ (a), $N(t,L)$ (b), and $R(t,L)$ (c). In each plot, only data for $L\ge 16384$ are shown. The straight lines indicate power laws in the central (scaling) region.

Figure 12

Log-log plots of rescaled lifetime moments $\langle T^k\rangle /L^{(\delta +k-1)z}$ of boundary-driven avalanches, with $k=1,2,$ and 3. Apart from finite-size corrections and statistical fluctuations, the data are supposed to fall on straight horizontal lines.

Figure 13

Log-log plot of ${\rho }_{a,\infty }$ against $\langle z\rangle -z_c$. The straight line indicates the power law predicted by the scaling theory. The best fit would indeed be obtained with an exponent $0.243\left(5\right)$. The leftmost point was obtained by simulating 40 lattices with $L={10}^6$ for $5\times {10}^7$ time steps.

Figure 14

Log-log plot of $\langle s\rangle$ against $z_c-\langle z\rangle$ for subcritical avalanches in the FES version. The upper straight line indicates the power law predicted by the scaling theory. The best fit would indeed be obtained with an exponent $2.68\left(2\right)$. The lower straight line shows the behavior that would have been expected, if the Oslo model were in the DP universality class. The leftmost point was obtained by simulating 3600 avalanches on a lattice with $L=5\times {10}^5$.

Figure 15

(a) Log-log plot of $\langle s\rangle$ (lowest triple of curves) and $\langle s^k\rangle /\langle s^{k-1}\rangle$ for $k=2$ and 3 (topmost triples) against $L$. Each curve in each triple corresponds to a different periodic start configuration (periods 101, 273, and 86 from top to bottom), and each value of $L$ is a multiple of the corresponding period. The data are divided by $L^2$ which makes the central curve in the lowest triple horizontal and makes the central curves in the two upper triples scale as $L^{1/4}$ (straight line). (b) Analogous results size weighted for moments of avalanche lifetimes. This time only data for the two smaller values of $\langle z\rangle$ are shown, and all data are divided by powers of $L$ which make the critical curve of each pair horizontal.

Figure 16

Log-log plots of $P_t(t,L)$ (a), $N(t,L)$ ( b), and $R(t,L)$ (c). In each plot, the raw data were divided by the conjectured power laws, i.e., the actually plotted data are $t^{7/40}{P}_t(t,L)$, $t^{-2/5}N\left(t\right)$, and $t^{-7/10}R\left(t\right)$. As in Fig. 17, $R\left(t\right)$ is the rms distance from the first toppling. In each panel, the straight lines indicate the error bars mentioned in the text.

Figure 17

Log-log plot of the avalanche size distributions for the fixed-energy version at $\langle z\rangle =1.732601$, and for different lattice sizes which are all multiples of 273. The actually plotted data are $s^{10/9}P\left(s\right)$.

Figure 18

Log-log plots of rescaled survival probabilities $t^{7/40}{P}_t\left(t\right)$, if the evolution of each still surviving avalanche is terminated at time $t_{\max }$. Here, the largest lattices (for the largest $t_{\max }$) had $L=2^{21}$. The average stress was $1.732591$ (i.e., slightly subcritical), but it was verified that results were indistinguishable at slightly supercritical $\langle z\rangle$. From our previous simulations we would have expected the curves to become horizontal for large $t$ and large $t_{\max }$.

Figure 19

Log-log plot of variances of the total stress on intervals of length $2^k,{\mathrm{Var}}_k=\mathrm{Var}\left[Z_k\right]$ with $Z_k={\sum }_{i=i_0}^{i_0+k-1\mathrm{mod}L}{z}_i$. Each curve was obtained by terminating the evolution of avalanches at $t_{\max }$, and the total CPU time used for each curve was roughly the same. Lattice sizes were up to $4\times {10}^6$, and $\langle z\rangle$ is very close to critical.

Figure 20

Main plot: log-log plot of ${\rho }_a\left(t\right)$ for $\langle z\rangle =359/208=1.732596$ and $L=524140$. For the upper curve (scheme A) each run started from a periodic configuration, while the lower curve (scheme B) used for each run the final state of the previous run, with half of the sites with $z_i=2$ declared as unstable. Each curve is based on 508 runs. The inset shows the same data multiplied by $t^{1/8}$. They coincide for $t>{10}^4$ within statistical errors. The straight lines in the inset indicate the error $\pm 0.002$ of $\theta$.

Figure 21

Activity densities for supercritical FES systems, large enough to have negligible finite-size effects. Initial configurations were chosen according to scheme A. The numbers in the legend indicate $\langle z\rangle$ (top curve has largest $\langle z\rangle$).

Figure 22

(a) Activity densities for near critical FES systems. Initial conditions used scheme A, and lattices are large enough to have negligible finite-size effects. The numbers in the legend indicate $\langle z\rangle$ (top curve has largest $\langle z\rangle$). (b) Data collapse obtained by plotting $t^{1/8}{\rho }_a\left(t\right)$ against $(z_c-\langle z\rangle )t^{1/{\nu }_t}$ with ${\nu }_t=40/21$. There are altogether 27 curves overlaid in this plot, for $\langle z\rangle$ ranging from 1.7143 to 1.7391.

Figure 23

Log-log plot of $s^{10/9}P(s,L)$ against $s/L^{9/4}$ for bulk-driven open systems. In (a) the driving is uniformly distributed over the entire region $[1,L]$, while in (b) the center sites are driven. In both cases, we see huge violations of scaling, which are biggest for uniform driving.

Figure 24

Log-log plot of $P_t(t,L)$ (a), $N(t,L)$ (b), and $R(t,L)$ (c) for center-driven systems, similar to Fig. 11 for boundary-driven avalanches. The straight lines indicate power laws in the central (scaling) region.

Figure 25

Roughness $W$ of interfaces constructed from the FES Oslo model, on lattices with density $\langle z\rangle =473/273=1.732601$ and lattice sizes which are multiples of 273. (a) Shows the raw data, while (b) shows a data collapse obtained by plotting $L^{-{\alpha }_{\mathrm{qEW}}}W$ against $t/L^z$. In this and the following two figures, averages are taken over all runs, those which are still active at time $t$ and those which had already died. If we had excluded the latter from the averages, interfaces would be slightly less rough (roughness increases sharply immediately before interfaces get pinned), but this would not affect their scaling (while it would affect the scaling of average height).

Figure 26

(a) Log-log plot of local squared roughnesses of interfaces constructed from the FES Oslo model, on lattices with density $\langle z\rangle =473/273=1.732601$ and size $L=131040$. The straight lines indicate the anomalous roughening power laws. (b) Shows the same data, but divided by $d^2$. According Eq. (46) one should see a data collapse for $d\ll t^{1/z}\ll L$. There is indeed a perfect collapse for $t\approx {10}^6$, if the largest distances $d$ are discarded. Panel (c), finally, is analogous to (a) but shows data for the alternative interface definition that is based on number of topplings instead of received units of stress.