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The sand-pile model and Tutte polynomials

2003
*
Advances in Applied Mathematics
*

A new explicit bijection between spanning trees

doi:10.1016/s0196-8858(02)00524-9
fatcat:hzgzb2iw5bcudcd5wlwxrxg2wm
*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] . ...##
###
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

1993
*
Physica A: Statistical Mechanics and its Applications
*

For AA

doi:10.1016/0378-4371(93)90267-8
fatcat:xzo7m3a5tvevxopwjfry6butje
*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] . ...##
###
Stability-to-instability transition in the structure of large-scale networks

2012
*
Physical Review E
*

Given

doi:10.1103/physreve.86.066106
pmid:23368003
fatcat:aarsvkj55nhuzpk77fsjarjfx4
*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. ...##
###
Multiparking Functions, Graph Searching, and the Tutte Polynomial
[article]

2006
*
arXiv
*
pre-print

In general

arXiv:math/0607602v1
fatcat:qfhsio6qyzg3jc34oyd267u5ri
*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. ...##
###
Multiparking functions, graph searching, and the Tutte polynomial

2008
*
Advances in Applied Mathematics
*

In general

doi:10.1016/j.aam.2007.03.001
fatcat:lzo4jes6bbfjxmbjjmuo2ytyf4
*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. ...##
###
On the abelianity of the stochastic sandpile model
[article]

2016
*
arXiv
*
pre-print

We consider a stochastic variant of

arXiv:1607.05561v1
fatcat:audzyvpkqvf5jiafdy3hva35vy
*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. ...##
###
The Looping Rate and Sandpile Density of Planar Graphs

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. ...

##
###
A chip-firing game and Dirichlet eigenvalues

2002
*
Discrete Mathematics
*

We also examine

doi:10.1016/s0012-365x(02)00434-x
fatcat:cy4yrqp7bzamrblhi25zme6kua
*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] ). ...##
###
Primer for the algebraic geometry of sandpiles
[article]

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. ...

##
###
Learning about critical phenomena from scribbles and sandpiles

2015
*
ESAIM Proceedings and Surveys
*

This text grew out of

doi:10.1051/proc/201551004
fatcat:lswmbfnf4fbzrbuus2z7ebfbca
*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*). ...##
###
Multiple and inverse topplings in the Abelian Sandpile Model

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 ...

##
###
Sandpile models
[article]

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. ...

##
###
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]

2022
*
arXiv
*
pre-print

(I already know

arXiv:2101.04168v3
fatcat:kh5eurusrjevbkttpuq24p3s4q
*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. ...
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