Stability of patterns in the Abelian sandpile

Abstract

We show that the patterns in the Abelian sandpile are stable. The proof combines the structure theory for the patterns with the regularity machinery for non-divergence form elliptic equations. The stability results allows one to improve weak-* convergence of the Abelian sandpile to pattern convergence for certain classes of solutions.

16 pages, 4 figures Corrected rate of growth in Theorem 1 item (1); other minor corrections