# Critical Density of the Abelian Manna Model via a Multi-type Branching Process

## TL;DR

This paper introduces a multi-type branching process to accurately determine the critical density of the Abelian Manna model, improving upon mean-field theories by incorporating key spatio-temporal correlations.

## Contribution

The authors develop an analytical method that captures essential correlations in the Abelian Manna model, providing more accurate critical density estimates across dimensions.

## Key findings

- Analytical expression for critical density in arbitrary dimensions.
- Improved accuracy over mean-field theories.
- Method validated against numerical simulation data.

## Abstract

A multi-type branching process is introduced to mimic the evolution of the avalanche activity and determine the critical density of the Abelian Manna model. This branching process incorporates partially the spatio-temporal correlations of the activity, which are essential for the dynamics, in particular in low dimensions. An analytical expression for the critical density in arbitrary dimensions is derived, which significantly improves the results over mean-field theories, as confirmed by comparison to the literature on numerical estimates from simulations. The method can easily be extended to lattices and dynamics other than those studied in the present work.