Sandpiles on the Square Lattice
Abstrak
AbstractWe give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice $${\mathbb{Z}^2}$$Z2 . We also determine the asymptotic spectral gap, asymptotic mixing time, and prove a cutoff phenomenon for the recurrent state abelian sandpile model on the torus $${\left(\mathbb{Z}/m\mathbb{Z}\right)^2}$$Z/mZ2 . The techniques use analysis of the space of functions on $${\mathbb{Z}^2}$$Z2 which are harmonic modulo 1. In the course of our arguments, we characterize the harmonic modulo 1 functions in $${\ell^p(\mathbb{Z}^2)}$$ℓp(Z2) as linear combinations of certain discrete derivatives of Green’s functions, extending a result of Schmidt and Verbitskiy (Commun Math Phys 292(3):721–759, 2009. arXiv:0901.3124 [math.DS]).
Topik & Kata Kunci
Penulis (3)
Robert D. Hough
D. Jerison
Lionel Levine
Akses Cepat
- Tahun Terbit
- 2017
- Bahasa
- en
- Total Sitasi
- 19×
- Sumber Database
- Semantic Scholar
- DOI
- 10.1007/s00220-019-03408-5
- Akses
- Open Access ✓