Directed Abelian sandpile with multiple downward neighbors

Area of Science:

• Statistical Physics
• Complex Systems
• Dynamical Processes

Background:

• The directed Abelian sandpile model is a fundamental model in statistical physics.
• Previous solutions were limited to the K=2 case on a square lattice.
• Avalanche clusters in the K=2 case are known to be compact.

Purpose of the Study:

• To extend the exact solution of the directed Abelian sandpile model to cases with K>2 downward neighbors.
• To analyze the structural differences and similarities of avalanche clusters for K>2 compared to K=2.
• To investigate the critical exponents and large-scale behavior of the model for K>2.

Main Methods:

• Exact solution of the directed Abelian sandpile model for K=3.
• Analysis of avalanche cluster structures for K>2.
• Calculation and comparison of critical exponents for K=2 and K>2.

Main Results:

• The K=3 case of the directed Abelian sandpile model on a square lattice is solved exactly.
• For K>2, avalanche clusters exhibit complex structures with holes and side branches, unlike the compact clusters in the K=2 case.
• Despite structural differences, critical exponents for K>2 are identical to those of the K=2 case.
• The large-scale structure of avalanches for K>2 demonstrates a tendency to converge towards the K=2 case.

Conclusions:

• The exact solution for the directed Abelian sandpile model is successfully extended to K>2.
• Qualitative differences in avalanche cluster morphology for K>2 do not alter the critical exponents.
• The model exhibits universal behavior in its critical exponents, independent of certain structural complexities.