11 research outputs found
Abelian networks II. Halting on all inputs
Abelian networks are systems of communicating automata satisfying a local
commutativity condition. We show that a finite irreducible abelian network
halts on all inputs if and only if all eigenvalues of its production matrix lie
in the open unit disk.Comment: Supersedes sections 5 and 6 of arXiv:1309.3445v1. To appear in
Selecta Mathematic
Absorbing-state phase transition and activated random walks with unbounded capacities
In this article, we study the existence of an absorbing-state phase transition of an Abelian process that generalises the Activated Random Walk (ARW). Given a vertex transitive G=(V,E), we associate to each site x∈V a capacity wx≥0, which describes how many inactive particles x can hold, where {wx}x∈V is a collection of i.i.d random variables. When G is an amenable graph, we prove that if E[wx]0. Moreover, in the former case, we provide bounds for the critical density that match the ones available in the classical Activated Random Walk
Abelian networks III. The critical group
The critical group of an abelian network is a finite abelian group that
governs the behavior of the network on large inputs. It generalizes the
sandpile group of a graph. We show that the critical group of an irreducible
abelian network acts freely and transitively on recurrent states of the
network. We exhibit the critical group as a quotient of a free abelian group by
a subgroup containing the image of the Laplacian, with equality in the case
that the network is rectangular. We generalize Dhar's burning algorithm to
abelian networks, and estimate the running time of an abelian network on an
arbitrary input up to a constant additive error.Comment: supersedes sections 7 and 8 of arXiv:1309.3445v1. To appear in the
Journal of Algebraic Combinatoric
Uniform threshold for fixation of the stochastic sandpile model on the line
We consider the abelian stochastic sandpile model. In this model, a site is
deemed unstable when it contains more than one particle. Each unstable site,
independently, is toppled at rate , sending two of its particles to
neighbouring sites chosen independently. We show that when the initial average
density is less than , the system locally fixates almost surely. We
achieve this bound by analysing the parity of the total number of times each
site is visited by a large number of particles under the sandpile dynamics.Comment: 21 pages including bibliography, 19 withou
Abelian networks IV. Dynamics of nonhalting networks
An abelian network is a collection of communicating automata whose state
transitions and message passing each satisfy a local commutativity condition.
This paper is a continuation of the abelian networks series of Bond and Levine
(2016), for which we extend the theory of abelian networks that halt on all
inputs to networks that can run forever. A nonhalting abelian network can be
realized as a discrete dynamical system in many different ways, depending on
the update order. We show that certain features of the dynamics, such as
minimal period length, have intrinsic definitions that do not require
specifying an update order.
We give an intrinsic definition of the \emph{torsion group} of a finite
irreducible (halting or nonhalting) abelian network, and show that it coincides
with the critical group of Bond and Levine (2016) if the network is halting. We
show that the torsion group acts freely on the set of invertible recurrent
components of the trajectory digraph, and identify when this action is
transitive.
This perspective leads to new results even in the classical case of sinkless
rotor networks (deterministic analogues of random walks). In Holroyd et. al
(2008) it was shown that the recurrent configurations of a sinkless rotor
network with just one chip are precisely the unicycles (spanning subgraphs with
a unique oriented cycle, with the chip on the cycle). We generalize this result
to abelian mobile agent networks with any number of chips. We give formulas for
generating series such as where is the number of recurrent chip-and-rotor configurations with
chips; is the diagonal matrix of outdegrees, and is the adjacency
matrix. A consequence is that the sequence completely
determines the spectrum of the simple random walk on the network.Comment: 95 pages, 21 figure