61 research outputs found
Sandpile probabilities on triangular and hexagonal lattices
We consider the Abelian sandpile model on triangular and hexagonal lattices.
We compute several height probabilities on the full plane and on half-planes,
and discuss some properties of the universality of the model.Comment: 26 pages, 12 figures. v2 and v3: minor correction
The looping rate and sandpile density of planar graphs
We give a simple formula for the looping rate of loop-erased random walk on a
finite planar graph. The looping rate is closely related to the expected amount
of sand in a recurrent sandpile on the graph. The looping rate formula is
well-suited to taking limits where the graph tends to an infinite lattice, and
we use it to give an elementary derivation of the (previously computed) looping
rate and sandpile densities of the square, triangular, and honeycomb lattices,
and compute (for the first time) the looping rate and sandpile densities of
many other lattices, such as the kagome lattice, the dice lattice, and the
truncated hexagonal lattice (for which the values are all rational), and the
square-octagon lattice (for which it is transcendental)
Studying Self-Organized Criticality with Exactly Solved Models
This is a somewhat expanded version of the notes of a series of lectures
given at Lausanne and Stellenbosch in 1998-99. They are intended to provide a
pedagogical introduction to the abelian sandpile model of self-organized
criticality, and its related models : the q=0 state Potts model, Takayasu
aggregation model, the voter model, spanning trees, Eulerian walkers model etc.
It provides an overview of the known results, and explains the equivalence of
these models. Some open questions are discussed in the concluding section.Comment: Latex with epsf, 47 pages, 14 figure
The abelian sandpile and related models
The Abelian sandpile model is the simplest analytically tractable model of
self-organized criticality. This paper presents a brief review of known results
about the model. The abelian group structure allows an exact calculation of
many of its properties. In particular, one can calculate all the critical
exponents for the directed model in all dimensions. For the undirected case,
the model is related to q= 0 Potts model. This enables exact calculation of
some exponents in two dimensions, and there are some conjectures about others.
We also discuss a generalization of the model to a network of communicating
reactive processors. This includes sandpile models with stochastic toppling
rules as a special case. We also consider a non-abelian stochastic variant,
which lies in a different universality class, related to directed percolation.Comment: Typos and minor errors fixed and some references adde
Critical Behavior of the Sandpile Model as a Self-Organized Branching Process
Kinetic equations, which explicitly take into account the branching nature of
sandpile avalanches, are derived. The dynamics of the sandpile model is
described by the generating functions of a branching process. Having used the
results obtained the renormalization group approach to the critical behavior of
the sandpile model is generalized in order to calculate both critical exponents
and height probabilities.Comment: REVTeX, twocolumn, 4 page
Multipoint correlators in the Abelian sandpile model
We revisit the calculation of height correlations in the two-dimensional
Abelian sandpile model by taking advantage of a technique developed recently by
Kenyon and Wilson. The formalism requires to equip the usual graph Laplacian,
ubiquitous in the context of cycle-rooted spanning forests, with a complex
connection. In the case at hand, the connection is constant and localized along
a semi-infinite defect line (zipper). In the appropriate limit of a trivial
connection, it allows one to count spanning forests whose components contain
prescribed sites, which are of direct relevance for height correlations in the
sandpile model. Using this technique, we first rederive known 1- and 2-site
lattice correlators on the plane and upper half-plane, more efficiently than
what has been done so far. We also compute explicitly the (new) next-to-leading
order in the distances ( for 1-site on the upper half-plane,
for 2-site on the plane). We extend these results by computing new correlators
involving one arbitrary height and a few heights 1 on the plane and upper
half-plane, for the open and closed boundary conditions. We examine our lattice
results from the conformal point of view, and confirm the full consistency with
the specific features currently conjectured to be present in the associated
logarithmic conformal field theory.Comment: 60 pages, 21 figures. v2: reformulation of the grove theorem, minor
correction
Explicit characterization of the identity configuration in an Abelian Sandpile Model
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 Z^2 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.Comment: 11 pages, 8 figure
Higher Order and boundary Scaling Fields in the Abelian Sandpile Model
The Abelian Sandpile Model (ASM) is a paradigm of self-organized criticality
(SOC) which is related to conformal field theory. The conformal fields
corresponding to some height clusters have been suggested before. Here we
derive the first corrections to such fields, in a field theoretical approach,
when the lattice parameter is non-vanishing and consider them in the presence
of a boundary.Comment: 7 pages, no figure
Exact solutions for a mean-field Abelian sandpile
We introduce a model for a sandpile, with N sites, critical height N and each
site connected to every other site. It is thus a mean-field model in the
spin-glass sense. We find an exact solution for the steady state probability
distribution of avalanche sizes, and discuss its asymptotics for large N.Comment: 10 pages, LaTe
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