Critical density of the Abelian Manna model via a multitype branching process

A multitype branching process is introduced to mimic the evolution of the avalanche activity and determine the critical density of the Abelian Manna model. This branching process incorporates partially the spatiotemporal correlations of the activity, which are essential for the dynamics, in particular in low dimensions. An analytical expression for the critical density in arbitrary dimensions is derived, which significantly improves the results over mean-field theories, as confirmed by comparison to the literature on numerical estimates from simulations. The method can easily be extended to lattices and dynamics other than those studied in the present work.

Figure 1

The six different active motives of the one-dimensional AMM. The central site carries either three (${\alpha }_i$ for $i=0,1,2$) or two particles (${\beta }_i$ for $i=0,1,2$) and is to topple in the next microscopic time step. The occupation number of sites neighboring the central site are representative only in as far as their parity is concerned, with $i$ indicating the number of sites that carry an odd number. Their occupation number is posttoppling (after they may have toppled themselves, but prior to the central site toppling), whereas the central site is shown pretopping (before it topples). The configurations differ in whether neighboring sites may or may not topple themselves due to being charged by the toppling site. Although we may picture the configurations being situated on a one-dimensional lattice, their spatial orientation is irrelevant, and so we do not distinguish left and right neighbors.

Figure 2

Example of a toppling on a one-dimensional lattice. The initial state ${\alpha }_1$ goes over into state ${\beta }_2$ with probability (w.p.) $\zeta /4$, which is the joint probability of three independent events: the independent toppling of two particles to one side (w.p. $1/4$) and the occupation of a next-nearest -neighboring site (w.p. $\zeta$). The latter assumption ignores spatial correlations.

Figure 3

Estimate of the asymptotic density in the fixed-energy version of the AMM on random 5-regular graphs of different sizes. The approximate critical point, identified as the onset of asymptotic survival, at around ${\zeta }_c=0.62$ is very close to the theoretical prediction ${\zeta }_c=1/2+1/q=0.6$.