267 research outputs found
The Lattice structure of Chip Firing Games and Related Models
In this paper, we study a famous discrete dynamical system, the Chip Firing
Game, used as a model in physics, economics and computer science. We use order
theory and show that the set of reachable states (i.e. the configuration space)
of such a system started in any configuration is a lattice, which implies
strong structural properties. The lattice structure of the configuration space
of a dynamical system is of great interest since it implies convergence (and
more) if the configuration space is finite. If it is infinite, this property
implies another kind of convergence: all the configurations reachable from two
given configurations are reachable from their infimum. In other words, there is
a unique first configuration which is reachable from two given configurations.
Moreover, the Chip Firing Game is a very general model, and we show how known
models can be encoded as Chip Firing Games, and how some results about them can
be deduced from this paper. Finally, we define a new model, which is a
generalization of the Chip Firing Game, and about which many interesting
questions arise.Comment: See http://www.liafa.jussieu.fr/~latap
Monomials, Binomials, and Riemann-Roch
The Riemann-Roch theorem on a graph G is related to Alexander duality in
combinatorial commutive algebra. We study the lattice ideal given by chip
firing on G and the initial ideal whose standard monomials are the G-parking
functions. When G is a saturated graph, these ideals are generic and the Scarf
complex is a minimal free resolution. Otherwise, syzygies are obtained by
degeneration. We also develop a self-contained Riemann-Roch theory for artinian
monomial ideals.Comment: 18 pages, 2 figures, Minor revision
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
Root system chip-firing I: Interval-firing
Jim Propp recently introduced a variant of chip-firing on a line where the
chips are given distinct integer labels. Hopkins, McConville, and Propp showed
that this process is confluent from some (but not all) initial configurations
of chips. We recast their set-up in terms of root systems: labeled chip-firing
can be seen as a root-firing process which allows the moves for whenever
, where is the set of
positive roots of a root system of Type A and is a weight of this
root system. We are thus motivated to study the exact same root-firing process
for an arbitrary root system. Actually, this central root-firing process is the
subject of a sequel to this paper. In the present paper, we instead study the
interval root-firing processes determined by for
whenever or , for any . We prove that these interval-firing processes are always confluent,
from any initial weight. We also show that there is a natural way to
consistently label the stable points of these interval-firing processes across
all values of so that the number of weights with given stabilization is a
polynomial in . We conjecture that these Ehrhart-like polynomials have
nonnegative integer coefficients.Comment: 54 pages, 12 figures, 2 tables; v2: major revisions to improve
exposition; v3: to appear in Mathematische Zeitschrift (Math. Z.
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