13 research outputs found
Algebraic and combinatorial aspects of sandpile monoids on directed graphs
The sandpile group of a graph is a well-studied object that combines ideas
from algebraic graph theory, group theory, dynamical systems, and statistical
physics. A graph's sandpile group is part of a larger algebraic structure on
the graph, known as its sandpile monoid. Most of the work on sandpiles so far
has focused on the sandpile group rather than the sandpile monoid of a graph,
and has also assumed the underlying graph to be undirected. A notable exception
is the recent work of Babai and Toumpakari, which builds up the theory of
sandpile monoids on directed graphs from scratch and provides many connections
between the combinatorics of a graph and the algebraic aspects of its sandpile
monoid.
In this paper we primarily consider sandpile monoids on directed graphs, and
we extend the existing theory in four main ways. First, we give a combinatorial
classification of the maximal subgroups of a sandpile monoid on a directed
graph in terms of the sandpile groups of certain easily-identifiable subgraphs.
Second, we point out certain sandpile results for undirected graphs that are
really results for sandpile monoids on directed graphs that contain exactly two
idempotents. Third, we give a new algebraic constraint that sandpile monoids
must satisfy and exhibit two infinite families of monoids that cannot be
realized as sandpile monoids on any graph. Finally, we give an explicit
combinatorial description of the sandpile group identity for every graph in a
family of directed graphs which generalizes the family of (undirected)
distance-regular graphs. This family includes many other graphs of interest,
including iterated wheels, regular trees, and regular tournaments.Comment: v2: Cleaner presentation, new results in final section. Accepted for
publication in J. Combin. Theory Ser. A. 21 pages, 5 figure
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
Markov chains, -trivial monoids and representation theory
We develop a general theory of Markov chains realizable as random walks on
-trivial monoids. It provides explicit and simple formulas for the
eigenvalues of the transition matrix, for multiplicities of the eigenvalues via
M\"obius inversion along a lattice, a condition for diagonalizability of the
transition matrix and some techniques for bounding the mixing time. In
addition, we discuss several examples, such as Toom-Tsetlin models, an exchange
walk for finite Coxeter groups, as well as examples previously studied by the
authors, such as nonabelian sandpile models and the promotion Markov chain on
posets. Many of these examples can be viewed as random walks on quotients of
free tree monoids, a new class of monoids whose combinatorics we develop.Comment: Dedicated to Stuart Margolis on the occasion of his sixtieth
birthday; 71 pages; final version to appear in IJA
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
Random walks on rings and modules
We consider two natural models of random walks on a module over a finite
commutative ring driven simultaneously by addition of random elements in
, and multiplication by random elements in . In the coin-toss walk,
either one of the two operations is performed depending on the flip of a coin.
In the affine walk, random elements are sampled
independently, and the current state is taken to . For both models,
we obtain the complete spectrum of the transition matrix from the
representation theory of the monoid of all affine maps on under a suitable
hypothesis on the measure on (the measure on can be arbitrary).Comment: 26 pages, 1 figure, minor improvements, final versio
Local-to-Global Principles for the Hitting Sequence of a Rotor Walk
In rotor walk on a finite directed graph, the exits from each vertex follow a prescribed periodic sequence. Here we consider the case of rotor walk where a particle starts from a designated source vertex and continues until it hits a designated target set, at which point the walk is restarted from the source. We show that the sequence of successively hit targets, which is easily seen to be eventually periodic, is in fact periodic. We show moreover that reversing the periodic patterns of all rotor sequences causes the periodic pattern of the hitting sequence to be reversed as well. The proofs involve a new notion of equivalence of rotor configurations, and an extension of rotor walk incorporating time-reversed particles.Massachusetts Institute of Technology. Undergraduate Research Opportunities ProgramNational Science Foundation (U.S.) (Grant 0644877)National Science Foundation (U.S.) (Grant 1001905)National Science Foundation (U.S.) (Postdoctoral Research Fellowship)National Science Foundation (U.S.). Research Experience for Undergraduates (Program
Deterministic Abelian Sandpile Models and Patterns
In this thesis we want to study the ASM in connection with its capability to produce interesting patterns. it is a surprising example of model that shows the emergence of patterns but maintains the property of being analytically tractable. Then it is qualitatively different from other typical growth models --like Eden model, the diffusion limit aggregation, or the surface deposition -- indeed while in these models the growth of the patterns is confined on the surfaces and the inner structures, once formed, are frozen and do not evolve anymore, in the ASM the patterns formed grow in size but at the same time the internal structures aquire structure, as it has been noted in several papers.
There have been several earlier studies of the spatial patterns in sandpile models. The first of them was by Liu et.al.
The asymptotic shape of the boundaries of the patterns produced in centrally seeded sandpile model on different periodic backgrounds was
discussed in a work of Dhar of 1999. Borgne et.al. obtained bounds on the rate of growth of these boundaries, and later these bounds were improved by Fey et.al. and Levine et.al.
An analysis of different periodic structures found in the patterns were first carried out by Ostojic who also first noted the exact quadratic nature of the toppling function within a patch.
Wilson et.al. have developed a very efficient algorithm to generate patterns for a large numbers of particles added, which allows them to generate pictures of patterns with N up to 2^26.
There are other models, which are related to the Abelian Sandpile Model,e.g., the Internal Diffusion-Limited Aggregation (IDLA), Eulerian walkers (also called the rotor-router model), and the infinitely-divisible sandpile, which also show similar structure.
For the IDLA, Gravner and Quastel showed that the asymptotic shape of the growth pattern is related to the classical Stefan problem in hydrodynamics, and determined the exact radius of the pattern with a single point source.
Levine and Peres have studied patterns with multiple sources in these models, and proved the existence of a limit shape. Limiting shapes for the non-Abelian sandpile has recently been studied by Fey et.al.
The results of our investigation toward a comprehension of the patterns emerging in the ASM are reported along the thesis.
In chapter 3 we will introduce some new algebraic operators, and in addition to , over the space of the sandpile configurations, that will be in the following basic ingredients in the creation of patterns in the sandpile. We derive some Temperley-Lieb like relations they satisfy. At the end of the chapter we show how do they are closely related to multitopplings and which consequences has that relation on the action of on recurrent configurations.
In chapter 4 we search for a closed formula to characterize the Identity configuration of the ASM. At this scope we study the ASM on the square lattice, in different geometries, and in a variant with directed edges, the F-lattice or pseudo-Manhattan lattice. Cylinders, through their extra symmetry, allow an easy characterization of the identity which is a homogeneous function. In the directed version, the pseudo-Manhattan lattice, we see a remarkable exact self-similar structure at different sizes, which results in the possibility to give a closed formula for the identity, this work has been published.
In chapter 5 we reach the cardinal point of our study, here we present the theory of strings and patches. The regions of a configuration periodic in space, called patches, are the ingredients of pattern formation.
In a last paper of Dhar, a condition on the shape of patch interfaces has been established, and proven at a coarse-grained level. We discuss how this result is strengthened by avoiding the coarsening, and describe the emerging fine-level structures, including linear interfaces and
rigid domain walls with a residual one-dimensional translational
invariance. These structures, that we shall call strings, are
macroscopically extended in their periodic direction, while showing
thickness in a full range of scales between the microscopic lattice
spacing and the macroscopic volume size.
We first explore the relations among these objects and then we present full classification of them, which leads to the construction and explanation of a Sierpinski triangular structure, which displays patterns of all the possible patches
Proceedings of JAC 2010. Journées Automates Cellulaires
The second Symposium on Cellular Automata “Journ´ees Automates Cellulaires” (JAC 2010) took place in Turku, Finland, on December 15-17, 2010. The first two conference days were held in the Educarium building of the University of Turku, while the talks of the third day were given onboard passenger ferry boats in the beautiful Turku archipelago, along the route Turku–Mariehamn–Turku. The conference was organized by FUNDIM, the Fundamentals of Computing and Discrete Mathematics research center at the mathematics department of the University of Turku.
The program of the conference included 17 submitted papers that were selected by the international program committee, based on three peer reviews of each paper. These papers form the core of these proceedings. I want to thank the members of the program committee and the external referees for the excellent work that have done in choosing the papers to be presented in the conference. In addition to the submitted papers, the program of JAC 2010 included four distinguished invited speakers: Michel Coornaert (Universit´e de Strasbourg, France), Bruno Durand (Universit´e de Provence, Marseille, France), Dora Giammarresi (Universit` a di Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨at Gie_en, Germany). I sincerely thank the invited speakers for accepting our invitation to come and give a plenary talk in the conference. The invited talk by Bruno Durand was eventually given by his co-author Alexander Shen, and I thank him for accepting to make the presentation with a short notice. Abstracts or extended abstracts of the invited presentations appear in the first part of this volume.
The program also included several informal presentations describing very recent developments and ongoing research projects. I wish to thank all the speakers for their contribution to the success of the symposium. I also would like to thank the sponsors and our collaborators: the Finnish Academy of Science and Letters, the French National Research Agency project EMC (ANR-09-BLAN-0164), Turku Centre for Computer Science, the University of Turku, and Centro Hotel. Finally, I sincerely thank the members of the local organizing committee for making the conference possible.
These proceedings are published both in an electronic format and in print. The electronic proceedings are available on the electronic repository HAL, managed by several French research agencies. The printed version is published in the general publications series of TUCS, Turku Centre for Computer Science. We thank both HAL and TUCS for accepting to publish the proceedings.Siirretty Doriast