Proof of breaking of self-organized criticality in a nonconservative Abelian sandpile model

By computer simulations, it was reported that the Bak-Tang-Wiesenfeld (BTW) model loses self-organized criticality (SOC) when some particles are annihilated in a toppling process in the bulk of system. We give a rigorous proof that the BTW model loses SOC as soon as the annihilation rate becomes positive. To prove this, a nonconservative Abelian sandpile model is defined on a square lattice, which has a parameter α (≥ 1) representing the degree of breaking of the conservation law. This model is reduced to be the BTW model when α = 1. By calculating the average number of topplings in an avalanche 〈T〉 exactly, it is shown that for any α > 1, 〈T〉 11(r) is also calculated analytically and we show that C11(r) is bounded by an exponential function when α > 1, although C11(r) ∼ r-2d was proved by Majumdar and Dhar for the d-dimensional BTW model. A critical exponent ν11 characterizing the divergence of the correlation length ξ as α → 1 is defined as ξ ∼ |α - 1|-ν11 and our result gives an upper bound ν11 ≤ 1/2. ©2000 The American Physical Society.