Explicit characterization of the identity configuration in an Abelian sandpile model

2008/09/30 by Sergio Caracciolo, Guglielmo Paoletti, Andrea Sportiello
Computer Science · Mathematics · Physics and Astronomy · #Abelian group #Abelian sandpile model #Advanced Mathematical Modeling in Engineering #Homogeneous #Lattice (music) #Nonlinear Photonic Systems #Square (algebra) #Square lattice #Theoretical and Computational Physics #cond-mat.stat-mech #hep-lat #math-ph #math.MP #nlin.AO

paper · pdf · doi:10.1088/1751-8113/41/49/495003

published as J.Phys.A41:495003,2008 · 11 pages, 8 figures

arxiv created 2008/10/05 · openalex publication_date 2008/10/29 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

Since the work of Creutz, identifying the group identities for the Abelian sandpile model (ASM) on a given lattice is a puzzling issue: on rectangular portions of complex quasi-self-similar structures arise. We study the ASM on the square lattice, in different geometries, and a variant with directed edges. Cylinders, through their extra symmetry, allow an easy determination of the identity, which is a homogeneous function. The directed variant on square geometry shows a remarkable exact structure, asymptotically self-similar.

Citations

• The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
• Determination of the exponent for SAWs on the two-dimensional Manhattan lattice
• Logarithmic scaling for height variables in the Abelian sandpile model
• The Abelian sandpile and related models
• Percolation, quantum tunnelling and the integer Hall effect