Sandpile patterns on a regular graph of degree eight

Abstract

The system is in critical condition if even a small disturbance can lead to global changes.
These are, for example, any phase transitions: in water cooled to zero degrees, one crystallization
center rapidly grows to a large cluster. The concept of self-organizing criticality was first proposed by Back, Tang and Weisenfeld in 1987. In their work, they described a system that has become a classic model of self-organizing criticality: on a square grid, in some nodes, there are grains of sand, a finite number in total. If there are more than three grains of sand in one of the nodes, a toppling occurs: four grains of sand from this node are redistributed to neighboring nodes, this can cause topplings in them, then in their neighbors... Collapses will occur in an
avalanche-like manner until the system returns to an equilibrium state, this process is called relaxation.
This article presents the results of an experimental and theoretical study of the following problem. Consider a regular graph whose vertices are points in the plane, both coordinates of which are integers, and each vertex is connected to the 8 nearest vertices. Put a large number of grains of sand at the point (0,0) and relax. The relaxation result has an obvious fractal structure, visible in computer experiments, and parts of this structure can be described.
We classify some emerging patterns and propose hypotheses about their structure (based on similar results for other regular graphs). Estimates for the average number of sand in the emerging patterns are proved.