Mean-field behavior of the sandpile model below the upper critical dimension

TL;DR

This study investigates the critical behavior of the sandpile model across dimensions 2 to 6, revealing deviations from mean-field theory in four dimensions and suggesting an upper critical dimension of at least five.

Contribution

The paper provides large-scale numerical simulations of the sandpile model, analyzing how dissipation and lattice size affect critical exponents and identifying the upper critical dimension.

Findings

Scaling exponents in 4D differ from mean-field predictions

Upper critical dimension is at least 5

Some exponents follow mean-field behavior below the upper critical dimension

Abstract

We present results of large scale numerical simulations of the Bak, Tang and Wiesenfeld sandpile model. We analyze the critical behavior of the model in Euclidean dimensions $2\leq d\leq 6$. We consider a dissipative generalization of the model and study the avalanche size and duration distributions for different values of the lattice size and dissipation. We find that the scaling exponents in $d=4$ significantly differ from mean-field predictions, thus suggesting an upper critical dimension $d_c\geq 5$. Using the relations among the dissipation rate $\epsilon$ and the finite lattice size $L$, we find that a subset of the exponents displays mean-field values below the upper critical dimensions. This behavior is explained in terms of conservation laws.