100 research outputs found
Results and conjectures on the Sandpile Identity on a lattice
International audienceIn this paper we study the identity of the Abelian Sandpile Model on a rectangular lattice.This configuration can be computed with the burning algorithm, which, starting from the empty lattice, computes a sequence of configurations, the last of which is the identity.We extend this algorithm to an infinite lattice, which allows us to prove that the first steps of the algorithm on a finite lattice are the same whatever its size.Finally we introduce a new configuration, which shares the intriguing properties of the identity, but is easier to study
Height variables in the Abelian sandpile model: scaling fields and correlations
We compute the lattice 1-site probabilities, on the upper half-plane, of the
four height variables in the two-dimensional Abelian sandpile model. We find
their exact scaling form when the insertion point is far from the boundary, and
when the boundary is either open or closed. Comparing with the predictions of a
logarithmic conformal theory with central charge c=-2, we find a full
compatibility with the following field assignments: the heights 2, 3 and 4
behave like (an unusual realization of) the logarithmic partner of a primary
field with scaling dimension 2, the primary field itself being associated with
the height 1 variable. Finite size corrections are also computed and
successfully compared with numerical simulations. Relying on these field
assignments, we formulate a conjecture for the scaling form of the lattice
2-point correlations of the height variables on the plane, which remain as yet
unknown. The way conformal invariance is realized in this system points to a
local field theory with c=-2 which is different from the triplet theory.Comment: 68 pages, 17 figures; v2: published version (minor corrections, one
comment added
Apollonian structure in the Abelian sandpile
The Abelian sandpile process evolves configurations of chips on the integer
lattice by toppling any vertex with at least 4 chips, distributing one of its
chips to each of its 4 neighbors. When begun from a large stack of chips, the
terminal state of the sandpile has a curious fractal structure which has
remained unexplained. Using a characterization of the quadratic growths
attainable by integer-superharmonic functions, we prove that the sandpile PDE
recently shown to characterize the scaling limit of the sandpile admits certain
fractal solutions, giving a precise mathematical perspective on the fractal
nature of the sandpile.Comment: 27 Pages, 7 Figure
Sandpile models
This survey is an extended version of lectures given at the Cornell
Probability Summer School 2013. The fundamental facts about the Abelian
sandpile model on a finite graph and its connections to related models are
presented. We discuss exactly computable results via Majumdar and Dhar's
method. The main ideas of Priezzhev's computation of the height probabilities
in 2D are also presented, including explicit error estimates involved in
passing to the limit of the infinite lattice. We also discuss various questions
arising on infinite graphs, such as convergence to a sandpile measure, and
stabilizability of infinite configurations.Comment: 72 pages - v3 incorporates referee's comments. References closely
related to the lectures were added/update
Directed nonabelian sandpile models on trees
We define two general classes of nonabelian sandpile models on directed trees
(or arborescences) as models of nonequilibrium statistical phenomena. These
models have the property that sand grains can enter only through specified
reservoirs, unlike the well-known abelian sandpile model.
In the Trickle-down sandpile model, sand grains are allowed to move one at a
time. For this model, we show that the stationary distribution is of product
form. In the Landslide sandpile model, all the grains at a vertex topple at
once, and here we prove formulas for all eigenvalues, their multiplicities, and
the rate of convergence to stationarity. The proofs use wreath products and the
representation theory of monoids.Comment: 43 pages, 5 figures; introduction improve
Identity Configurations of the Sandpile Group
The abelian sandpile model on a connected graph yields a finite abelian group Q of recurrent configurations which is closely related to the combinatorial Laplacian. We consider the identity configuration of the sandpile group on graphs with large edge multiplicities, called “thick” graphs. We explicitly compute the identity configuration for all thick paths using a recursion formula. We then analyze the thick cycle and explicitly compute the identity configuration for the three-cycle, the four-cycle, and certain types of symmetric cycles. The latter is a special case of a more general symmetry theorem we prove that applies to an arbitrary graph.https://digitalcommons.imsa.edu/sci_dsw/1001/thumbnail.jp
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