Open AccessThe sandpile identity element on an ellipse
Author(s) -
Andrew Melchionna
Publication year - 2022
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2022029
Subject(s) - ellipse , lattice (music) , identity (music) , combinatorics , abelian sandpile model , element (criminal law) , physics , square lattice , graph , geometry , mathematics , condensed matter physics , statistical physics , acoustics , political science , law , ising model
We consider certain elliptical subsets of the square lattice. The recurrent representative of the identity element of the sandpile group on this graph consists predominantly of a biperiodic pattern, along with some noise. We show that as the lattice spacing tends to 0, the fraction of the area taken up by the pattern in the identity element tends to 1.