Exact integration of height probabilities in the Abelian Sandpile Model

Abstract

The height probabilities for the recurrent configurations in the Abelian Sandpile Model on the square lattice have analytic expressions, in terms of multidimensional quadratures. At first, these quantities have been evaluated numerically with high accuracy, and conjectured to be certain cubic rational-coefficient polynomials in 1/pi. Later their values have been determined by different methods. We revert to the direct derivation of these probabilities, by computing analytically the corresponding integrals. Yet another time, we confirm the predictions on the probabilities, and thus, as a corollary, the conjecture on the average height.

arXiv comment17 pages, added references

References in corpus (17)

• Height variables in the Abelian sandpile model: scaling fields and correlations
• Introduction to the Sandpile Model
• Coordinate space methods for the evaluation of Feynman diagrams in lattice field theory
• Logarithmic scaling for height variables in the Abelian sandpile model
• Lattice Perturbation Theory for $O(N)$-Symmetric $\sigma$-Models with General Nearest-Neighbour Action I. Conventional Perturbation Theory
• Boundary height fields in the Abelian sandpile model
• Four-Loop Perturbative Expansion for the Lattice $N$-Vector Model
• Driving sandpiles to criticality and beyond
• Pre-logarithmic and logarithmic fields in a sandpile model
• Abelian Sandpile Model on the Honeycomb Lattice
• The approach to criticality in sandpiles
• Algebraic algorithm for the computation of one-loop Feynman diagrams in lattice QCD with Wilson fermions
• Return probability for the loop-erased random walk and mean height in sandpile : a proof
• Multiple and inverse topplings in the Abelian Sandpile Model
• Logarithmic two-point correlators in the Abelian sandpile model
• Renormalization flow for unrooted forests on a triangular lattice
• Application of a coordinate space method for the evaluation of lattice Feynman diagrams in two dimensions

Cited by in corpus (7)

• 25 Years of Self-Organized Criticality: Concepts and Controversies
• Feedback mechanisms for self-organization to the edge of a phase transition
• Logarithmic conformal invariance in the Abelian sandpile model
• Sandpile models in the large
• Sandpile probabilities on triangular and hexagonal lattices
• Correlations in the $n\to 0$ limit of the dense O(n) loop model
• Multipoint correlators in the Abelian sandpile model