Height Probabilities in the Abelian Sandpile Model on the Generalized Trees

Abstract

This paper deals with the Abelian sandpile model on the generalized trees with certain given boundary condition. Using a combinatorial method, we obtain the exact expressions for all single-site probabilities and some two-site joint probabilities. Also, we prove that the sites near the boundary have a different height probability from those away from it in bulk for the Bethe lattice with the boundary condition, which is the same as those results found by Grassberger and Manna [Some more sandpiles,” J.Phys.(France)\(51,1077-1098(1990)\)] and proved by Haiyan chen and Fuji Zhang [“Height probabilities in the Abelian sandpile on the generalized finite Bethe lattice” J. Math. Phys. \(54, 083503 (2013))\).