Particle-density fluctuations and universality in the conserved stochastic sandpile

We examine fluctuations in particle density in the restricted-height, conserved stochastic sandpile (CSS). In this and related models, the global particle density is a temperaturelike control parameter. Thus local fluctuations in this density correspond to disorder; if this disorder is a relevant perturbation of directed percolation (DP), then the CSS should exhibit non-DP critical behavior. We analyze the scaling of the variance $V_{\ell }$ of the number of particles in regions of ${\ell }^d$ sites in extensive simulations of the quasistationary state in one and two dimensions. Our results, combined with a Harris-like argument for the relevance of particle-density fluctuations, strongly suggest that conserved stochastic sandpiles belong to a universality class distinct from that of DP.

Figure 1

Stationary correlation function $\left|h\right(\ell \left)\right|$ in the one-dimensional CSS at criticality for system sizes $L=2000$, 5000, 10 000, 20 000, 40 000, and 80 000. Upper inset: $\left|h\right(\ell \left)\right|$ in the one-dimensional critical CSS, system size $L=20000$, for random ($\times$), natural (points), and uniform (circles) initial configurations. Lower inset: $\left|h\right(\ell \left)\right|$ in the two-dimensional CSS at criticality, system sizes $L=128,256,...,4096$.

Figure 2

Particle-number variance $V_{\ell }$ over strips of size $\ell$ in the one-dimensional CSS at criticality. The curves have been shifted by subtracting $V_1$ from each result. System sizes as in Fig. 1. Inset: particle-number variance $V_{\ell }$ over squares of side $\ell$ in the two-dimensional CSS at the critical point, system sizes $L=128,256,...,4096$. Curves have been shifted by subtracting $V_1$ from each result. The dashed lines correspond to the power laws, $V_{\ell }\sim {\ell }^{\varphi }$, reported in [27], with $\varphi =0.575\left(25\right)$ in one dimension and $\varphi =1.55\left(3\right)$ in 2D.

Figure 3

Scaling plot of particle-number variance in the one-dimensional CSS at criticality using scaling variables $V^{*}\equiv V_{\ell }/L^{0.483}$ and $x\equiv \ell /L$. System sizes (lower to upper at left) $L=20000, 40000$, and $80000$. Upper inset: scaling plot for the two-dimensional CSS, using $V^{*}\equiv V_{\ell }/L^{1.60}$, with $L=512$, 1024, 2048, and 4096. Lower inset: effective growth exponent $\varphi \left(x\right)$ versus $x$ in one dimension. The dashed line denotes the value of $\varphi$ above which particle-density fluctuations are relevant.