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The sand-pile model and Tutte polynomials

Robert Cori, Yvan Le Borgne
2003 Advances in Applied Mathematics  
A new explicit bijection between spanning trees and recurrent configurations of the sand-pile model is given.  ...  It gives a bijective proof of a result of Merino López expressing the generating function of recurrent configurations as an evaluation of the Tutte polynomial.  2003 Elsevier Science (USA).  ...  Introduction The sand-pile model was introduced as a model in physics [1, 6, 7] and was also considered by combinatorialists as the chip firing game [2] [3] [4] [5] .  ... 
doi:10.1016/s0196-8858(02)00524-9 fatcat:hzgzb2iw5bcudcd5wlwxrxg2wm

Page 2754 of Mathematical Reviews Vol. , Issue 2004d [page]

2004 Mathematical Reviews  
; Talence) The sand-pile model and Tutte polynomials.  ...  The authors describe an explicit bijection between spanning trees of a connected rooted multigraph G and recurrent configurations on G in the sand-pile model (a model in physics, related to the chip-firing  ... 

Abelian avalanches and Tutte polynomials

Andrei Gabrielov
1993 Physica A: Statistical Mechanics and its Applications  
For AA model on a graph, statistics of avalanches is linked to Tutte polynomials associated with this graph and its subgraphs.  ...  An important property of this Abelian sand pile (ASP) model is the presence of an Abelian (commutative) group governing its dynamics.  ...  "Tutte polynomials" [14] .  ... 
doi:10.1016/0378-4371(93)90267-8 fatcat:xzo7m3a5tvevxopwjfry6butje

Stability-to-instability transition in the structure of large-scale networks

Dandan Hu, Peter Ronhovde, Zohar Nussinov
2012 Physical Review E  
Given the relation between Tutte and Jones polynomials, our results further suggest a link between the above complexity transitions and transitions associated with random knots.  ...  We further employ multivariate Tutte polynomials to show that increasing q emulates increasing T for a general Potts model, leading to a similar stability region at low T.  ...  Tran, and L. Zdeborová for discussions and ongoing work.  ... 
doi:10.1103/physreve.86.066106 pmid:23368003 fatcat:aarsvkj55nhuzpk77fsjarjfx4

Multiparking Functions, Graph Searching, and the Tutte Polynomial [article]

Dimitrije Kostic, Catherine Yan
2006 arXiv   pre-print
In general the polynomial that encodes information about subgraphs of G is the Tutte polynomial, which is the generating function for two parameters, namely the internal and external activities, associated  ...  In particular, the bijection induced by the breadth-first search leads to a new characterization of external activity, and hence a representation of Tutte polynomial by the reversed sum of G-multiparking  ...  We also thank Ira Gessel for helpful comments on Tutte polynomials, and for sharing the unpublished portion of a preprint of [9] with us.  ... 
arXiv:math/0607602v1 fatcat:qfhsio6qyzg3jc34oyd267u5ri

Multiparking functions, graph searching, and the Tutte polynomial

Dimitrije Kostić, Catherine H. Yan
2008 Advances in Applied Mathematics  
In general the polynomial that encodes information about subgraphs of G is the Tutte polynomial t G (x, y), which is the generating function for two parameters, namely the internal and external activities  ...  These many valuations make the Tutte polynomial one of the most fundamental tools in algebraic graph theory.  ...  We also thank Ira Gessel for helpful comments on Tutte polynomials, and for sharing the unpublished portion of a preprint of [9] with us.  ... 
doi:10.1016/j.aam.2007.03.001 fatcat:lzo4jes6bbfjxmbjjmuo2ytyf4

On the abelianity of the stochastic sandpile model [article]

François Nunzi
2016 arXiv   pre-print
We consider a stochastic variant of the Abelian Sandpile Model (ASM) on a finite graph, introduced by Chan, Marckert and Selig.  ...  As a corollary, we show that the stationary distribution still does not depend on how sand grains are added onto the graph in our model, answering a conjecture of Selig.  ...  Acknowledgments The author would like to thank Andrea Sportiello for sharing his vision and pertinent questions, and Jérémie Bouttier for his help.  ... 
arXiv:1607.05561v1 fatcat:audzyvpkqvf5jiafdy3hva35vy

The Looping Rate and Sandpile Density of Planar Graphs

Adrien Kassel, David B. Wilson
2016 The American mathematical monthly  
The looping rate is closely related to the expected amount of sand in a recurrent sandpile on the graph.  ...  densities of the square, triangular, and honeycomb lattices, and compute (for the first time) the looping rate and sandpile densities of many other lattices, such as the kagome lattice, the dice lattice  ...  Le Borgne, The sand-pile model and Tutte polynomials, Adv. in Appl. Math. 30 (2003), 44–52. 9. D. Dhar, Self-organized critical state of sandpile automaton models, Phys. Rev.  ... 
doi:10.4169/amer.math.monthly.123.1.19 fatcat:ohrq6shikjezfhp22fj4zyvu3q

A chip-firing game and Dirichlet eigenvalues

Fan Chung, Robert B. Ellis
2002 Discrete Mathematics  
We also examine the relations among three equinumerous families, the set of spanning forests on S with roots in the boundary of S, a set of "critical" configurations of chips, and a coset group, called  ...  the sandpile group associated with S.  ...  The sandpile group and rooted forests The sandpile group originated in the study of modeling the behavior of grains of sand piled onto the nodes of a structure ( [14, 15] ).  ... 
doi:10.1016/s0012-365x(02)00434-x fatcat:cy4yrqp7bzamrblhi25zme6kua

Primer for the algebraic geometry of sandpiles [article]

David Perkinson, Jacob Perlman, John Wilmes
2011 arXiv   pre-print
The Abelian Sandpile Model (ASM) is a game played on a graph realizing the dynamics implicit in the discrete Laplacian matrix of the graph.  ...  An extended summary of the ASM and of the required algebraic geometry is provided.  ...  As the name suggests, we think of a configuration c as a pile of sand on the nonsink vertices of G having c v grains of sand at vertex v.  ... 
arXiv:1112.6163v2 fatcat:hg5rqb7k3zbgdm33d74r42f4yi

Learning about critical phenomena from scribbles and sandpiles

Adrien Kassel, Aurélien Garivier
2015 ESAIM Proceedings and Surveys  
This text grew out of the notes for a talk given at the Journées MAS 2014.  ...  We give a concise report on our recent work on loop-erased random walk and related processes (spanning trees, cycle-rooted spanning forests, and sandpiles) on planar lattices and on graphs embedded in  ...  sand, although the model is of course not intended as a realistic model of piles of sand).  ... 
doi:10.1051/proc/201551004 fatcat:lswmbfnf4fbzrbuus2z7ebfbca

Multiple and inverse topplings in the Abelian Sandpile Model

S. Caracciolo, G. Paoletti, A. Sportiello
2012 The European Physical Journal Special Topics  
The Abelian Sandpile Model is a cellular automaton whose discrete dynamics reaches an out-of-equilibrium steady state resembling avalanches in piles of sand.  ...  We prove a number of algebraic properties of this monoid, and describe their practical implications on the emerging structures of the model.  ...  of the graph (not surprisingly, as the Tutte polynomial of a graph is a reformulation of the Potts Model partition function, the adjacency between variables being encoded by the graph, see e.g  ... 
doi:10.1140/epjst/e2012-01652-9 fatcat:yhgzzweorngizp2wqg4ci77ziy

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.  ...  The main ideas of Priezzhev's computation of the height probabilities in 2D are also presented, including explicit error estimates involved in passing to the limit of the infinite lattice.  ...  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

Page 479 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
ISBN 0-521-81722-6 (Julian Cole) 2004f:00005 00A30 (03A05, 03B35, 68T01) Cori, Robert (with Le Borgne, Yvan) The sand-pile model and Tutte polynomials.  ...  (Summary) 2004j:52021 52C35 (05B35, 14F40) — (with Forge, David) A note on Tutte polynomials and Orlik-Solomon algebras. (English summary) European J. Combin. 24 (2003), no. 8, 1081-1087.  ... 

This Week's Finds in Mathematical Physics (1-50) [article]

John C. Baez
2022 arXiv   pre-print
(I already know the cover page looks weird).  ...  These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995.  ...  The gauge-invariant operators and hamiltonian are realized in a Hilbert space of open path and loop functionals.  ... 
arXiv:2101.04168v3 fatcat:kh5eurusrjevbkttpuq24p3s4q
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