Logarithmic corrections of the avalanche distributions of sandpile models at the upper critical dimension

TL;DR

This study investigates the avalanche distributions in sandpile models at the upper critical dimension, finding that four dimensions exhibit mean-field behavior with logarithmic corrections, supported by numerical and analytical results.

Contribution

It provides numerical evidence and analytical insights into the logarithmic corrections affecting avalanche distributions at the upper critical dimension of four.

Findings

Avalanche distributions in 4D follow mean-field exponents with logarithmic corrections.

Numerical data fits the logarithmic correction models well.

The upper critical dimension for the BTW model is identified as four.

Abstract

We study numerically the dynamical properties of the BTW model on a square lattice for various dimensions. The aim of this investigation is to determine the value of the upper critical dimension where the avalanche distributions are characterized by the mean-field exponents. Our results are consistent with the assumption that the scaling behavior of the four-dimensional BTW model is characterized by the mean-field exponents with additional logarithmic corrections. We benefit in our analysis from the exact solution of the directed BTW model at the upper critical dimension which allows to derive how logarithmic corrections affect the scaling behavior at the upper critical dimension. Similar logarithmic corrections forms fit the numerical data for the four-dimensional BTW model, strongly suggesting that the value of the upper critical dimension is four.