Dissipative Abelian Sandpile Models

Abstract

We introduce a family of abelian sandpile models with two parameters $n,m\in \mathbf{N}$ defined on finite lattices on $d$-dimensional torus. Sites with $2dn+m$ or more grains of sand are unstable and topple, and in each toppling $m$ grains dissipate from the system. Because of dissipation in bulk, the models are well-defined on the shift-invariant lattices and the infinite-volume limit of systems can be taken. From the determinantal expressions, we obtain the asymptotic forms of the avalanche propagators and the height-$(0,0)$ correlations of sandpiles for large distances in the infinite-volume limit in any dimensions $d\ge 2$. We show that both of them decay exponentially with the correlation length $$\xi (d,a)=(\sqrt{d}{\sinh }^{-1}\sqrt{a(a+2)}{)}^{-1},$$ if the dissipation rate $a=m/(2dn)$ is positive. By considering a series of models with increasing $n$, we discuss the limit $a\downarrow 0$ and the critical exponent defined by ${\nu }_a=-{\lim }_{a\downarrow 0}\log \xi (d,a)/\log a$ is determined as $${\nu }_a=1/2$$ for all $d\ge 2$. Comparison with the $q\downarrow 0$ limit of $q$-state Potts model in external magnetic field is discussed.

AMS-LaTeX, 33 pages, 6 figures

References in corpus (1)

• From elongated spanning trees to vicious random walks