4,318 research outputs found
A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees
Hereditary chip-firing models generalize the Abelian sandpile model and the
cluster firing model to an exponential family of games induced by covers of the
vertex set. This generalization retains some desirable properties, e.g.
stabilization is independent of firings chosen and each chip-firing equivalence
class contains a unique recurrent configuration. In this paper we present an
explicit bijection between the recurrent configurations of a hereditary
chip-firing model on a graph and its spanning trees.Comment: 13 page
Sandpiles and Dominos
We consider the subgroup of the abelian sandpile group of the grid graph
consisting of configurations of sand that are symmetric with respect to central
vertical and horizontal axes. We show that the size of this group is (i) the
number of domino tilings of a corresponding weighted rectangular checkerboard;
(ii) a product of special values of Chebyshev polynomials; and (iii) a
double-product whose factors are sums of squares of values of trigonometric
functions. We provide a new derivation of the formula due to Kasteleyn and to
Temperley and Fisher for counting the number of domino tilings of a 2m x 2n
rectangular checkerboard and a new way of counting the number of domino tilings
of a 2m x 2n checkerboard on a M\"obius strip.Comment: 35 pages, 24 figure
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
Introduction to the Sandpile Model
This article is based on a talk given by one of us (EVI) at the conference
``StatPhys-Taipei-1997''. It overviews the exact results in the theory of the
sandpile model and discusses shortly yet unsolved problem of calculation of
avalanche distribution exponents. The key ingredients include the analogy with
the critical reaction-diffusion system, the spanning tree representation of
height configurations and the decomposition of the avalanche process into waves
of topplings
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
Trees, parking functions, syzygies, and deformations of monomial ideals
For a graph G, we construct two algebras, whose dimensions are both equal to
the number of spanning trees of G. One of these algebras is the quotient of the
polynomial ring modulo certain monomial ideal, while the other is the quotient
of the polynomial ring modulo certain powers of linear forms. We describe the
set of monomials that forms a linear basis in each of these two algebras. The
basis elements correspond to G-parking functions that naturally came up in the
abelian sandpile model. These ideals are instances of the general class of
monotone monomial ideals and their deformations. We show that the Hilbert
series of a monotone monomial ideal is always bounded by the Hilbert series of
its deformation. Then we define an even more general class of monomial ideals
associated with posets and construct free resolutions for these ideals. In some
cases these resolutions coincide with Scarf resolutions. We prove several
formulas for Hilbert series of monotone monomial ideals and investigate when
they are equal to Hilbert series of deformations. In the appendix we discuss
the sandpile model.Comment: 33 pages; v2: appendix on sandpiles added, references added, typos
corrected; v3: references adde
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
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