7,671 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
On the complexity of the chip-firing reachability problem
In this paper, we study the complexity of the chip-firing reachability
problem. We show that for Eulerian digraphs, the reachability problem can be
decided in strongly polynomial time, even if the digraph has multiple edges. We
also show a special case when the reachability problem can be decided in
polynomial time for general digraphs: if the target distribution is recurrent
restricted to each strongly connected component. As a further positive result,
we show that the chip-firing reachability problem is in co-NP for general
digraphs. We also show that the chip-firing halting problem is in co-NP for
Eulerian digraphs
Conjugacy of Coxeter elements
For a Coxeter group (W,S), a permutation of the set S is called a Coxeter
word and the group element represented by the product is called a Coxeter
element. Moving the first letter to the end of the word is called a rotation
and two Coxeter elements are rotation equivalent if their words can be
transformed into each other through a sequence of rotations and legal
commutations.
We prove that Coxeter elements are conjugate if and only if they are rotation
equivalent. This was known for some special cases but not for Coxeter groups in
general
Riemann-Roch and Abel-Jacobi theory on a finite graph
It is well-known that a finite graph can be viewed, in many respects, as a
discrete analogue of a Riemann surface. In this paper, we pursue this analogy
further in the context of linear equivalence of divisors. In particular, we
formulate and prove a graph-theoretic analogue of the classical Riemann-Roch
theorem. We also prove several results, analogous to classical facts about
Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian.
As an application of our results, we characterize the existence or
non-existence of a winning strategy for a certain chip-firing game played on
the vertices of a graph.Comment: 35 pages. v3: Several minor changes made, mostly fixing typographical
errors. This is the final version, to appear in Adv. Mat
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