The Abelian Manna model on two fractal lattices

TL;DR

This study investigates the avalanche size distribution of the Abelian Manna model on two fractal lattices with the same dimension, revealing a generalized scaling law and a linear relationship among critical exponents and lattice properties.

Contribution

It demonstrates the generalization of the scaling law D(2−τ)=d_w to fractal lattices and uncovers a linear relationship among critical exponents and lattice dimensions.

Findings

The scaling law D(2−τ)=d_w holds for fractal lattices.

Critical exponents and lattice dimensions obey a linear relationship.

The lattice dimension, walk dimension, and critical exponent form a plane in parameter space.

Abstract

We analyze the avalanche size distribution of the Abelian Manna model on two different fractal lattices with the same dimension d_g=ln(3)/ln(2), with the aim to probe for scaling behavior and to study the systematic dependence of the critical exponents on the dimension and structure of the lattices. We show that the scaling law D(2-tau)=d_w generalizes the corresponding scaling law on regular lattices, in particular hypercubes, where d_w=2. Furthermore, we observe that the lattice dimension d_g, the fractal dimension of the random walk on the lattice d_w, and the critical exponent D, form a plane in 3D parameter space, i.e. they obey the linear relationship D=0.632(3) d_g + 0.98(1) d_w - 0.49(3).