Analytical approach to directed sandpile models on the Apollonian network

TL;DR

This paper provides an analytical study of directed sandpile models on the Apollonian network, deriving exact correlation functions and avalanche distributions for specific parameters, and analyzing flux properties through multifractal scaling.

Contribution

It introduces an analytical approach to directed sandpile models on the Apollonian network, including exact correlation functions and avalanche distributions for certain parameters.

Findings

Exact correlation functions for q=1,2

Analytical avalanche distributions in the infinite system limit

Multifractal analysis of local flux properties

Abstract

We investigate a set of directed sandpile models on the Apollonian network, which are inspired on the work by Dhar and Ramaswamy (PRL \textbf{63}, 1659 (1989)) for Euclidian lattices. They are characterized by a single parameter $q$, that restricts the number of neighbors receiving grains from a toppling node. Due to the geometry of the network, two and three point correlation functions are amenable to exact treatment, leading to analytical results for the avalanche distributions in the limit of an infinite system, for $q=1,2$. The exact recurrence expressions for the correlation functions are numerically iterated to obtain results for finite size systems, when larger values of $q$ are considered. Finally, a detailed description of the local flux properties is provided by a multifractal scaling analysis.