
Abelian Sandpile Models in Statistical Mechanics : Dissipative Abelian Sandpile ModelsAbelian Sandpile Models in Statistical Mechanics : Dissipative Abelian Sandpile Models 
"/Katori, Makoto/"Katori, Makoto
63pp.58

91 , 20150820 , Institute of Mathematics for Industry, Kyushu University
ISSN:21881200
Description
We introduce a family of abelian sandpile models with two parameters n, m ∈ N defined on finite lattices on ddimensional 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 welldefined on the shiftinvariant lattices and the infinitevolume 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 infinitevolume limit in any dimensions d ≥ 2. We show that both of them decay exponentially with the correlation length ξ(d, a) = (√<d> sinh^<−1>√<a(a + 2)> )^<−1>, if the dissipation rate a = m/(2dn) is positive. Considering a series of models with increasing n, we discuss the limit a ↓ 0 and the critical exponent defined by ν_a = −< lim>___<a↓0> (log ξ(d, a)) / (log a) is determined as ν_a = 1 / 2 for all d ≥ 2. Comparison with the q ↓ 0 limit of qstate Potts model in external magnetic field is discussed.
FullText
http://catalog.lib.kyushuu.ac.jp/handle/2324/1525727/p058.pdf