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Signed Chip Firing Games and symmetric Sandpile Models on the cycles

Robert Cori, Thi Ha Duong Phan, Thi Thu Huong Tran
2012 RAIRO - Theoretical Informatics and Applications  
We investigate the Sandpile Model and Chip Firing Game and an extension of these models on cycle graphs. The extended model consists of allowing a negative number of chips at each vertex.  ...  We give the characterization of reachable configurations and of fixed points of each model. At the end, we give explicit formula for the number of their fixed points.  ...  ) and their two generalizations called Symmetric Sandpile Model (SSPM) and signed chip firing game (SCFG).  ... 
doi:10.1051/ita/2012023 fatcat:7prnflccgre2rnqv6buuaycoty

On the sandpile model of modified wheels II

Zahid Raza, Mohammed M. M. Jaradat, Mohammed S. Bataineh, Faiz Ullah
2020 Open Mathematics  
We investigate the abelian sandpile group on modified wheels {\hat{W}}_{n} by using a variant of the dollar game as described in [N. L. Biggs, Chip-Firing and the critical group of a graph, J.  ...  More precisely, the sandpile group on {\hat{W}}_{n} is the direct product of two cyclic subgroups of order {a}_{n} and 3{a}_{n} for n even and of order {a}_{n} and 2{a}_{n} for n odd, respectively.  ...  Zahid Raza was supported by University of Sharjah under the grant number 1802144068 and MASEP research group.  ... 
doi:10.1515/math-2020-0094 fatcat:z5tpowghyjcnnoyno5t43xzv5i

Rotor-Routing Induces the Only Consistent Sandpile Torsor Structure on Plane Graphs [article]

Ankan Ganguly, Alex McDonough
2022 arXiv   pre-print
We conjecture that the Backman-Baker-Yuen algorithm is consistent, and that there are only three other consistent sandpile torsor algorithms on regular matroids, all with the same structure.  ...  We also define sandpile torsor algorithms on regular matroids and suggest a notion of consistency in this context.  ...  Finally, we thank Cyrus Peterpaul for many fruitful discussions on this topic.  ... 
arXiv:2203.15079v1 fatcat:acqu65wjlbgghm65uamvf2h2dy

The Sandpile Group of a Thick Cycle Graph [article]

Diane Christine Alar, Jonathan Celaya, Luis David García Puente, Micah Henson, Ashley K. Wheeler
2017 arXiv   pre-print
The majority of graphs whose sandpile groups are known are either regular or simple. We give an explicit formula for a family of non-regular multi-graphs called thick cycles.  ...  Its sandpile group is the direct sum of cyclic groups of orders given by quotients of greatest common divisors of minors of its Laplacian matrix.  ...  DMS-1156499) and the National Security Agency (grant No. H98230-116-1-0033) . Special thanks to onsite program director Dr. Suzanne Weekes of Worcester Polytechnic Institute.  ... 
arXiv:1710.06006v1 fatcat:nxocdcmjejdoxgthmtfg7akvae

Sandpile models [article]

Antal A. Járai
2018 arXiv   pre-print
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.  ...  We also discuss various questions arising on infinite graphs, such as convergence to a sandpile measure, and stabilizability of infinite configurations.  ...  Acknowledgements I thank Lionel Levine and Laurent Saloff-Coste for offering the opportunity to give a course at the Summer School.  ... 
arXiv:1401.0354v3 fatcat:jrob2t6akfgwpbufldguednfxa

Minimal free resolutions of the $G$-parking function ideal and the toppling ideal

Madhusudan Manjunath, Frank-Olaf Schreyer, John Wilmes
2014 Transactions of the American Mathematical Society  
It is an initial ideal of the toppling ideal I G , a lattice ideal intimately related to the chip-firing game on a graph. Both ideals were first studied by Cori, Rossin, and Salvy.  ...  A minimal free resolution for M G was given by Postnikov and Shapiro in the case when G is saturated, i.e., whenever there is at least one edge (u, v) for every ordered pair of distinct vertices u and  ...  We thank Farbod Shokrieh for stimulating discussions on potential theory and Gröbner bases. The first author thanks Matt Baker for the support and encouragement in the course of this work.  ... 
doi:10.1090/s0002-9947-2014-06248-x fatcat:zuxqlovxufa2noptw3cchh3xdu

Minimal Free Resolutions of the G-parking Function Ideal and the Toppling Ideal [article]

Madhusudan Manjunath, Frank-Olaf Schreyer, John Wilmes
2012 arXiv   pre-print
It is an initial ideal of the toppling ideal I_G, a lattice ideal intimately related to the chip-firing game on a graph. Both ideals were first studied by Cori, Rossin, and Salvy.  ...  A minimal free resolution for M_G was given by Postnikov and Shaprio in the case when G is saturated, i. e., whenever there is at least one edge (u,v) for every ordered pair of distinct vertices u and  ...  We thank Farbod Shokrieh for stimulating discussions on potential theory and Gröbner bases. The first author thanks Matt Baker for the support and encouragement in the course of this work.  ... 
arXiv:1210.7569v3 fatcat:a7ieinzyvfb2rlo2apesktysnm

G-parking functions, acyclic orientations and spanning trees

Brian Benson, Deeparnab Chakrabarty, Prasad Tetali
2010 Discrete Mathematics  
sandpile models and closely related chip-firing games.  ...  In this work, we study the properties of maximum G-parking functions and provide a new bijection between them and the set of spanning trees of G with no broken circuit.  ...  Acknowledgements The authors thank Matt Baker for introducing them to G-parking functions and for helpful discussions.  ... 
doi:10.1016/j.disc.2010.01.002 fatcat:hpzukienv5aopbxfdxe5d7cbwe

Abelian avalanches and Tutte polynomials

Andrei Gabrielov
1993 Physica A: Statistical Mechanics and its Applications  
In a non-dissipative case ( j ∆ ij = 0, for all i) an avalanche in the ASP model coincides with a chip-firing game on a graph [6] where ∆ is a Laplace matrix of the underlying graph.  ...  Abelian sandpiles were studied in [4], and one special case is treated in [5].  ...  More precisely, for any u ∈ R V + and v ∈ R V + , we have C u • C v = C u+v . (10) The proof (see Appendix B) follows arguments of [6] for chip-firing games.  ... 
doi:10.1016/0378-4371(93)90267-8 fatcat:xzo7m3a5tvevxopwjfry6butje

Toppleable Permutations, Excedances and Acyclic Orientations [article]

Arvind Ayyer, Daniel Hathcock, Prasad Tetali
2020 arXiv   pre-print
Inspired by the work of Hopkins, McConville and Propp (Elec. J.  ...  We conclude with observations on an extremal question of Cameron et al. concerning maximizers of (the number of) acyclic orientations, given a prescribed number of vertices and edges for the graph.  ...  Introduction The sandpile model on graphs has been extensively studied, ever since its introduction by Dhar [Dha90] and independently by Björner, Lovasz and Shor [BLS91] (where it is called the chip-firing  ... 
arXiv:2010.11236v2 fatcat:loqxcyumcvgsfbrg2x6etbx4gm

Toppleable permutations, excedances and acyclic orientations

Arvind Ayyer, Daniel Hathcock, Prasad Tetali
2022 Combinatorial Theory  
Inspired by the work of Hopkins, McConville and Propp (Elec. J.  ...  We conclude with observations on an extremal question of Cameron et al. concerning maximizers of (the number of) acyclic orientations, given a prescribed number of vertices and edges for the graph.  ...  Ravichandran for discussions on toppleable permutations while he was in residence at Institut Mittag-Leffler in Djursholm, Sweden during the semester of Spring, 2020.  ... 
doi:10.5070/c62156882 fatcat:eymnywqx5zczhgumkssy5ue52a

Simplicial and cellular trees [chapter]

Art M. Duval, Caroline J. Klivans, Jeremy L. Martin
2016 IMA Volumes in Mathematics and its Applications  
as the abelian sandpile model.  ...  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.  ...  Acknowledgements We thank our colleagues Amy Wagler and Jennifer Wagner for valuable suggestions regarding exposition and clarity.  ... 
doi:10.1007/978-3-319-24298-9_28 fatcat:oh5q37h55jcmvady7lukhisolq

Tiered trees and Theta operators [article]

Michele D'Adderio and Alessandro Iraci and Yvan LeBorgne and Marino Romero and Anna Vanden Wyngaerd
2022 arXiv   pre-print
In [Dugan-Glennon-Gunnells-Steingrimsson-2019], the authors introduce tiered trees to define combinatorial objects counting absolutely indecomposable representations of certain quivers, and torus orbits  ...  on certain homogeneous varieties.  ...  We refer to [21] for the basic theory of the sandpile model (also known as chip-firing game). Definition 7.7. Take G = (V, s, E) a graph with a distinguished vertex s called the sink.  ... 
arXiv:2202.05706v1 fatcat:ficbtb3tzbcmhe4fmi6bnxslsq

Woven Nematic Defects, Skyrmions and the Abelian Sandpile Model [article]

Thomas Machon, Gareth P. Alexander
2018 pre-print
model or chip-firing game.  ...  The physical correspondence between local Skyrmion flux and sandpile height is made between the two models.  ...  In statistical mechanics it indexes stable recurrent states in the Abelian sandpile model [32] or chip-firing games [33, 34] . The Abelian sandpile model (see Ref.  ... 
doi:10.1103/physrevlett.121.237801 arXiv:1808.02425v1 fatcat:fc5l4gzgdfed5pl7r2wtoqdkm4

Kadanoff Sand Pile Model. Avalanche Structure and Wave Shape [article]

Kevin Perrot, Eric Rémila
2013 arXiv   pre-print
Kadanoff et al inspire KSPM, extending the well known Sand Pile Model (SPM).  ...  Sand pile models are dynamical systems describing the evolution from N stacked grains to a stable configuration.  ...  Sand pile models are specializations of Chip Firing Games (CFG).  ... 
arXiv:1304.5109v1 fatcat:5szo6wr3pzhijdafghv6oq3tsm
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