%% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ %% Created for Nick Beaton at 2019-05-21 19:37:08 +1000 %% Saved with string encoding Unicode (UTF-8) @article{Aval:2013, Author = {Aval, Jean-Christophe and D'Adderio, Michele and Dukes, Mark and Hicks, Angela and Le Borgne, Yvan}, Booktitle = {Discrete Mathematics and Theoretical Computer Science}, Month = {01}, Title = {A q,t-analogue of Narayana numbers}, Year = {2013}} @article{Bak:1987, Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, Adsurl = {http://adsabs.harvard.edu/abs/1987PhRvL..59..381B}, Author = {{Bak}, P. and {Tang}, C. and {Wiesenfeld}, K.}, Doi = {10.1103/PhysRevLett.59.381}, Journal = {Physical Review Letters}, Keywords = {Degrees Of Freedom, Dynamical Systems, Flicker, Fractals, Noise Spectra, Self Organizing Systems, Arrays, Clumps, Strange Attractors, Systems Stability}, Month = jul, Pages = {381-384}, Title = {{Self-organized criticality - An explanation of 1/f noise}}, Volume = 59, Year = 1987, Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.59.381}} @article{Baker:2007, Author = {Baker, Matthew and Norin, Sergey}, Booktitle = {Advances in Mathematics}, Month = {11}, Pages = {766-788}, Title = {Riemann--Roch and Abel--Jacobi theory on a finite graph}, Volume = {215}, Year = {2007}} @article{Bernardi:2008, Author = {Olivier Bernardi}, Journal = {Electr. J. Comb.}, Title = {Tutte Polynomial, Subgraphs, Orientations and Sandpile Model: New Connections via Embeddings}, Volume = {15}, Year = {2008}} @article{Biggs:1997, Author = {L. Biggs, N}, Booktitle = {Bulletin of The London Mathematical Society - BULL LOND MATH SOC}, Month = {11}, Pages = {641-682}, Title = {Algebraic Potential Theory on Graphs}, Volume = {29}, Year = {1997}} @article{Biggs:1999, Author = {Biggs, N.L.}, Booktitle = {Journal of Algebraic Combinatorics}, Month = {01}, Pages = {25-45}, Title = {Chip-Firing and the Critical Group of a Graph}, Volume = {9}, Year = {1999}} @article{Bjorner1989165, Abstract = {We present two q-analogues of a hook length formula of Knuth for the number of linear extensions of a partially ordered set whose Hasse diagram is a rooted forest. These q-analogues give formulas for the inversion index and the major index generating functions over permutations which correspond to linear extensions of a labeled forest. They generalize and unify several other q-formulas appearing in the literature. For linear forests all of these formulas reduce to MacMahon's classical formula for ``q-counting'' multiset permutations according to the major index and inversion index. We also extend MacMahon's formula in another direction by q-counting all labelings of a fixed forest according to two very natural statistics on labeled forests which generalize the major index and inversion index on permutations. }, Author = {Anders Bj{\"o}rner and Michelle L Wachs}, Doi = {http://dx.doi.org/10.1016/0097-3165(89)90028-9}, Journal = {Journal of Combinatorial Theory, Series A}, Number = {2}, Pages = {165 - 187}, Title = {q-Hook length formulas for forests}, Url = {http://www.sciencedirect.com/science/article/pii/0097316589900289}, Volume = {52}, Year = {1989}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/0097316589900289}, Bdsk-Url-2 = {http://dx.doi.org/10.1016/0097-3165(89)90028-9}} @article{Bjorner:1991, Author = {Anders Bj{\"o}rner and L{\'a}szl{\'o} Lov{\'a}sz and Peter W. Shor}, Doi = {https://doi.org/10.1016/S0195-6698(13)80111-4}, Issn = {0195-6698}, Journal = {European Journal of Combinatorics}, Number = {4}, Pages = {283 - 291}, Title = {Chip-firing Games on Graphs}, Url = {http://www.sciencedirect.com/science/article/pii/S0195669813801114}, Volume = {12}, Year = {1991}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0195669813801114}, Bdsk-Url-2 = {https://doi.org/10.1016/S0195-6698(13)80111-4}} @article{Borwein1989291, Author = {David Borwein and Stuart Rankin and Lex Renner}, Doi = {http://dx.doi.org/10.1016/0012-365X(89)90272-0}, Journal = {Discrete Mathematics}, Number = {3}, Pages = {291 - 296}, Title = {Enumeration of injective partial transformations}, Url = {http://www.sciencedirect.com/science/article/pii/0012365X89902720}, Volume = {73}, Year = {1989}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/0012365X89902720}, Bdsk-Url-2 = {http://dx.doi.org/10.1016/0012-365X(89)90272-0}} @misc{OEIS, Author = {N. J. A. Sloane}, Note = {\url{http://oeis.org}}, Title = {The On-Line Encyclopedia of Integer Sequences}} @article{Bond:2016, Author = {Benjamin Bond and Lionel Levine}, Journal = {SIAM J. Discrete Math.}, Pages = {856-874}, Title = {Abelian Networks I. Foundations and Examples}, Volume = {30}, Year = {2016}} @article{BOUSQUETMELOU2010313, Author = {Mireille Bousquet-M{\'e}lou}, Doi = {http://dx.doi.org/10.1016/j.jcta.2009.10.001}, Journal = {Journal of Combinatorial Theory, Series A}, Keywords = {D-finite series}, Number = {3}, Pages = {313 - 344}, Title = {Families of prudent self-avoiding walks}, Url = {http://www.sciencedirect.com/science/article/pii/S0097316509001502}, Volume = {117}, Year = {2010}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0097316509001502}, Bdsk-Url-2 = {http://dx.doi.org/10.1016/j.jcta.2009.10.001}} @article{Bresenham1965, Acmid = {1663349}, Address = {Riverton, NJ, USA}, Author = {Bresenham, J. E.}, Doi = {10.1147/sj.41.0025}, Issn = {0018-8670}, Issue_Date = {March 1965}, Journal = {IBM Syst. J.}, Month = mar, Number = {1}, Numpages = {6}, Pages = {25--30}, Publisher = {IBM Corp.}, Title = {Algorithm for Computer Control of a Digital Plotter}, Url = {http://dx.doi.org/10.1147/sj.41.0025}, Volume = {4}, Year = {1965}, Bdsk-Url-1 = {http://dx.doi.org/10.1147/sj.41.0025}} @article{Bruhn:2011, Abstract = {It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary duals. In this paper we illustrate the new theory by exhibiting its implications for the cycle and bond matroids of infinite graphs. We also describe their algebraic cycle matroids, those whose circuits are the finite cycles and double rays, and determine their duals. Finally, we give a sufficient condition for a matroid to be representable in a sense adapted to infinite matroids. Which graphic matroids are representable in this sense remains an open question.}, Author = {Henning Bruhn and Reinhard Diestel}, Date-Modified = {2019-05-21 19:37:05 +1000}, Doi = {10.1016/j.disc.2010.12.015}, Issn = {0012-365X}, Journal = {Discrete Math.}, Keywords = {Infinite matroid, Graph, Duality, Whitney, Cycle matroid, Bond matroid, Representability}, Number = {15}, Pages = {1461 - 1471}, Title = {Infinite matroids in graphs}, Volume = {311}, Year = {2011}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0012365X10004693}, Bdsk-Url-2 = {https://doi.org/10.1016/j.disc.2010.12.015}} @article{Cara2015, Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, Adsurl = {http://adsabs.harvard.edu/abs/2015arXiv150806107C}, Archiveprefix = {arXiv}, Author = {{Caracciolo}, S. and {Paoletti}, G. and {Sportiello}, A.}, Eprint = {1508.06107}, Journal = {ArXiv e-prints}, Keywords = {Condensed Matter - Statistical Mechanics, High Energy Physics - Lattice, Mathematics - Combinatorics, Nonlinear Sciences - Cellular Automata and Lattice Gases}, Month = aug, Primaryclass = {cond-mat.stat-mech}, Title = {{Deterministic Abelian Sandpile and square-triangle tilings}}, Year = 2015} @article{Caracciolo:2011, Author = {Caracciolo, Sergio and Paoletti, Guglielmo and Sportiello, Andrea}, Booktitle = {The European Physical Journal Special Topics}, Month = {12}, Pages = {23--44}, Title = {Multiple and inverse topplings in the Abelian Sandpile Model}, Volume = {212}, Year = {2011}} @article{Cara2008, Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, Adsurl = {http://adsabs.harvard.edu/abs/2008JPhA...41W5003C}, Archiveprefix = {arXiv}, Author = {{Caracciolo}, S. and {Paoletti}, G. and {Sportiello}, A.}, Doi = {10.1088/1751-8113/41/49/495003}, Eid = {495003}, Eprint = {0809.3416}, Journal = {Journal of Physics A Mathematical General}, Month = dec, Pages = {5003}, Primaryclass = {cond-mat.stat-mech}, Title = {{Explicit characterization of the identity configuration in an Abelian sandpile model}}, Volume = 41, Year = 2008, Bdsk-Url-1 = {https://doi.org/10.1088/1751-8113/41/49/495003}} @article{Chen2013563, Author = {William Y.C. Chen and Oliver X.Q. Gao and Peter L. Guo}, Doi = {http://dx.doi.org/10.1016/j.aam.2013.05.001}, Issn = {0196-8858}, Journal = {Advances in Applied Mathematics}, Keywords = {( F , w ) -partition}, Number = {5}, Pages = {563 - 582}, Title = {q-Hook length formulas for signed labeled forests}, Url = {http://www.sciencedirect.com/science/article/pii/S0196885813000559}, Volume = {51}, Year = {2013}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0196885813000559}, Bdsk-Url-2 = {http://dx.doi.org/10.1016/j.aam.2013.05.001}} @article{Cori200344, Abstract = {A new explicit bijection between spanning trees and recurrent configurations of the sand-pile model is given. This mapping is such that the difference between the number of grains on a configuration and the external activity of the associate tree is the number of edges of the graph. It gives a bijective proof of a result of Merino L{\'o}pez expressing the generating function of recurrent configurations as an evaluation of the Tutte polynomial. }, Author = {Robert Cori and Yvan Le Borgne}, Date-Modified = {2019-05-21 19:05:09 +1000}, Doi = {10.1016/S0196-8858(02)00524-9}, Issn = {0196-8858}, Journal = {Adv. in Appl. Math.}, Number = {1--2}, Pages = {44 - 52}, Title = {The sand-pile model and Tutte polynomials}, Volume = {30}, Year = {2003}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0196885802005249}, Bdsk-Url-2 = {http://dx.doi.org/10.1016/S0196-8858(02)00524-9}} @misc{DeryckeYLB18, Author = {Henri Derycke and Yvan Le~Borgne}, Date-Modified = {2019-05-21 19:28:23 +1000}, Howpublished = {17e Journ{\'e}es Montoises d'Informatique Th{\'e}orique}, Title = {A definition and counting of biperiodic recurrent configurations in the sandpile model on $\mathbf{Z}^2$}, Url = {https://jm2018.sciencesconf.org/data/abstract_jm2018_Derycke.pdf}, Year = {2018}, Bdsk-Url-1 = {https://jm2018.sciencesconf.org/data/abstract_jm2018_Derycke.pdf}} @phdthesis{Derycke18, Author = {Derycke, Henri}, Date-Modified = {2019-05-21 19:06:51 +1000}, Hal_Id = {tel-01977878}, Hal_Version = {v1}, Keywords = {Sandpile model ; Recurrence ; Spanning tree ; Bootstrap percolation ; Q-analogue ; Tas de sable ; Arbre couvrant}, Note = {(French)}, Number = {2018BORD0327}, Pdf = {https://tel.archives-ouvertes.fr/tel-01977878/file/DERYCKE_HENRI_2018.pdf}, School = {{Universit{\'e} de Bordeaux}}, Title = {{Combinatorics on some stabilisations in the Abelian Sandpile Model on the square lattice Z${}^2$}}, Url = {https://tel.archives-ouvertes.fr/tel-01977878}, Year = {2018}, Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01977878}} @article{Dhar1995, Abstract = {The Abelian sandpile models feature a finite Abelian group G generated by the operators corresponding to particle addition at various sites. We study the canonical decomposition of G as a product of cyclic groups G=Z(d 1 )*Z(d 2 )*Z(d 3 )...*Z(d g ), where g is the least number of generators of G, and d i is a multiple of d i+1 . The structure of G is determined in terms of the toppling matrix Delta . We construct scalar functions, linear in the height variables of the pile, that are invariant under toppling at any site. These invariants provide convenient coordinates to label the recurrent configurations of the sandpile. For an L*L square lattice, we show that g=L. In this case, we observe that the system has non-trivial symmetries, transcending the obvious symmetries of the square, namely those coming from the action of the cyclotomic Galois group Gal L of the 2(L+1)th roots of unity (which operates on the set of eigenvalues of h). These eigenvalues are algebraic integers, the product of which is the order mod G mod . With the help of Gal L we are able to group the eigenvalues into certain subsets the products of which are separately integers, and thus obtain an explicit factorization of mod G mod . We also use Gal L to define other simpler sets of toppling invariants.}, Author = {D Dhar and P Ruelle and S Sen and D -N Verma}, Journal = {Journal of Physics A: Mathematical and General}, Number = {4}, Pages = {805}, Title = {Algebraic aspects of Abelian sandpile models}, Url = {http://stacks.iop.org/0305-4470/28/i=4/a=009}, Volume = {28}, Year = {1995}, Bdsk-Url-1 = {http://stacks.iop.org/0305-4470/28/i=4/a=009}} @article{Dhar1990, Author = {Dhar, Deepak}, Date-Modified = {2019-05-21 19:30:46 +1000}, Doi = {10.1103/PhysRevLett.64.1613}, Journal = {Phys. Rev. Lett.}, Number = {14}, Pages = {1613--1616}, Publisher = {American Physical Society}, Title = {Self-organized critical state of sandpile automaton models}, Volume = {64}, Year = {1990}, Bdsk-Url-1 = {http://link.aps.org/doi/10.1103/PhysRevLett.64.1613}, Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.64.1613}} @article{Dhar:2006, Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, Adsurl = {http://adsabs.harvard.edu/abs/2006PhyA..369...29D}, Author = {{Dhar}, D.}, Doi = {10.1016/j.physa.2006.04.004}, Journal = {Physica A Statistical Mechanics and its Applications}, Month = sep, Pages = {29-70}, Title = {{Theoretical studies of self-organized criticality}}, Volume = 369, Year = 2006, Bdsk-Url-1 = {https://doi.org/10.1016/j.physa.2006.04.004}} @article{dharSadhu:2013, Author = {Dhar, Deepak and Sadhu, Tridib}, Booktitle = {Journal of Statistical Mechanics: Theory and Experiment}, Month = {10}, Title = {A sandpile model for proportionate growth}, Volume = {2013}, Year = {2013}} @article{Dhar:2008, Author = {Dhar, Deepak and Sadhu, Tridib and Chandra, Samarth}, Booktitle = {EPL (Europhysics Letters)}, Month = {08}, Title = {Pattern formation in growing sandpiles}, Volume = {85}, Year = {2008}} @article{Dukes:2013, Author = {Mark Dukes and Yvan Le Borgne}, Doi = {https://doi.org/10.1016/j.jcta.2013.01.004}, Issn = {0097-3165}, Journal = {Journal of Combinatorial Theory, Series A}, Keywords = {Parallelogram polyomino, Sandpile model, Complete bipartite graph, Recurrent states, -Narayana polynomial}, Number = {4}, Pages = {816 - 842}, Title = {Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial}, Url = {http://www.sciencedirect.com/science/article/pii/S0097316513000150}, Volume = {120}, Year = {2013}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0097316513000150}, Bdsk-Url-2 = {https://doi.org/10.1016/j.jcta.2013.01.004}} @inproceedings{Elizalde:2009:SPS:1496770.1496778, Acmid = {1496778}, Address = {Philadelphia, PA, USA}, Author = {Elizalde, Sergi and Winkler, Peter}, Booktitle = {Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms}, Location = {New York}, Numpages = {8}, Pages = {68--75}, Title = {Sorting by Placement and Shift}, Url = {http://dl.acm.org/citation.cfm?id=1496770.1496778}, Year = {2009}, Bdsk-Url-1 = {http://dl.acm.org/citation.cfm?id=1496770.1496778}} @article{Foata:1978, Author = {Foata, Dominique and Sch{\"u}tzenberger, Marcel-Paul}, Doi = {10.1002/mana.19780830111}, Eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/mana.19780830111}, Journal = {Mathematische Nachrichten}, Number = {1}, Pages = {143-159}, Title = {Major Index and Inversion Number of Permutations}, Url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/mana.19780830111}, Volume = {83}, Year = {1978}, Bdsk-Url-1 = {https://onlinelibrary.wiley.com/doi/abs/10.1002/mana.19780830111}, Bdsk-Url-2 = {https://doi.org/10.1002/mana.19780830111}} @phdthesis{gamlin16, Abstract = {The focus of this thesis is to investigate the impact of the boundary conditions on configurations in the Abelian sandpile model. We have two main results to present in this thesis. Firstly we give a family of continuous, measure preserving, almost one-to-one mappings from the wired spanning forest to recurrent sandpiles. In the special case of \$Z{\^{ }}d\$, \$d {$\backslash$}geq 2\$, we show how these bijections yield a power law upper bound on the rate of convergence to the sandpile measure along any exhaustion of \$Z{\^{ }}d\$. Secondly we consider the Abelian sandpile on ladder graphs. For the ladder sandpile measure, \${$\backslash$}nu\$, a recurrent configuration on the boundary, I, and a cylinder event, E, we provide an upper bound for \${$\backslash$}nu(E{\ensuremath{|}}I) ? {$\backslash$}nu(E)\$.}, Author = {Samuel Gamlin}, Keywords = {abelian sandpile,spanning forest,wilson's algorithm,random walk}, Month = {April}, School = {University of Bath}, Title = {Boundary conditions in Abelian sandpiles}, Url = {http://opus.bath.ac.uk/50859/}, Year = {2016}, Bdsk-Url-1 = {http://opus.bath.ac.uk/50859/}} @article{gamlin2014, Author = {Gamlin, Samuel and J{\'a}rai, Antal}, Date-Modified = {2019-05-21 19:29:32 +1000}, Doi = {10.1214/EJP.v19-3542}, Fjournal = {Electronic Journal of Probability}, Journal = {Electron. J. Probab.}, Pages = {23 pp}, Pno = {117}, Publisher = {The Institute of Mathematical Statistics and the Bernoulli Society}, Title = {Anchored burning bijections on finite and infinite graphs}, Volume = {19}, Year = {2014}, Bdsk-Url-1 = {https://doi.org/10.1214/EJP.v19-3542}} @article{Garsia:2002, Acmid = {605323}, Address = {Amsterdam, The Netherlands, The Netherlands}, Author = {Garsia, A. M. and Haglund, J.}, Doi = {10.1016/S0012-365X(02)00343-6}, Issn = {0012-365X}, Issue_Date = {28 October 2002}, Journal = {Discrete Math.}, Month = oct, Number = {3}, Numpages = {41}, Pages = {677--717}, Publisher = {Elsevier Science Publishers B. V.}, Title = {A Proof of the Q, t-Catalan Positivity Conjecture}, Url = {http://dx.doi.org/10.1016/S0012-365X(02)00343-6}, Volume = {256}, Year = {2002}, Bdsk-Url-1 = {http://dx.doi.org/10.1016/S0012-365X(02)00343-6}} @article{Gravner2008, Author = {Janko Gravner and Alexander E. Holroyd}, Journal = {Electronic Journal of Probability}, Pages = {385--399}, Title = {Local Bootstrap Percolation}, Year = {2008}} @inproceedings{Haglund:2007, Author = {Jim Haglund}, Title = {The q , t-Catalan Numbers and the Space of Diagonal Harmonics with an Appendix on the Combinatorics of Macdonald Polynomials}, Year = {2007}} @misc{Holroyd03integrals, Author = {Alexander E. Holroyd and Thomas M. Liggett and Dan Romik}, Title = {Integrals, Partitions, and Cellular Automata}, Year = {2003}} @inbook{Holroyd2008, Abstract = {We give a rigorous and self-contained survey of the abelian sandpile model and otor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.}, Address = {Basel}, Author = {Holroyd, Alexander E. and Levine, Lionel and M{\'e}sz{\'a}ros, Karola and Peres, Yuyal and Propp, James and Wilson, David B.}, Booktitle = {In and Out of Equilibrium 2}, Doi = {10.1007/978-3-7643-8786-0_17}, Editor = {Sidoravicius, Vladas and Vares, Maria Eul{\'a}lia}, Isbn = {978-3-7643-8786-0}, Pages = {331--364}, Publisher = {Birkh{\"a}user Basel}, Title = {Chip-Firing and Rotor-Routing on Directed Graphs}, Url = {https://doi.org/10.1007/978-3-7643-8786-0_17}, Year = {2008}, Bdsk-Url-1 = {https://doi.org/10.1007/978-3-7643-8786-0_17}} @article{Hough:2017, Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, Adsurl = {http://adsabs.harvard.edu/abs/2017arXiv170300827H}, Archiveprefix = {arXiv}, Author = {{Hough}, B. and {Jerison}, D. and {Levine}, L.}, Eprint = {1703.00827}, Journal = {ArXiv e-prints}, Keywords = {Mathematics - Probability, 82C20, 60B15, 60J10}, Month = mar, Primaryclass = {math.PR}, Title = {{Sandpiles on the square lattice}}, Year = 2017} @article{Ivas1994, Abstract = {We investigate the properties of boundary avalanches in 2D Abelian sandpile model (ASM). We construct the one-to-one correspondence of boundary avalanches and two-rooted spanning trees. Using the connection between the obtained graph representation and lattice Green functions we calculate the exact values of critical exponents for size and lifetime distributions of avalanches starting at the open boundary that forms an angle alpha . We find that the probability of a boundary avalanche of the size s varies as s -1- pi 2 alpha / for large s and the probability of an avalanche of lifetime t varies as t -1-4 pi 5 alpha / for large t. The obtained values are verified by numerical simulations.}, Author = {E V Ivashkevich and D V Ktitarev and V B Priezzhev}, Journal = {Journal of Physics A: Mathematical and General}, Number = {16}, Pages = {L585}, Title = {Critical exponents for boundary avalanches in two-dimensional Abelian sandpile}, Url = {http://stacks.iop.org/0305-4470/27/i=16/a=004}, Volume = {27}, Year = {1994}, Bdsk-Url-1 = {http://stacks.iop.org/0305-4470/27/i=16/a=004}} @article{Jarai07, Author = {Antal A. J{\'a}rai and Russell Lyons}, Booktitle = {Markov Process. Related Fields 13}, Pages = {493--518}, Title = {Ladder sandpiles}, Year = {2007}} @article{Jarai:2014, Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, Adsurl = {http://adsabs.harvard.edu/abs/2014arXiv1401.0354J}, Archiveprefix = {arXiv}, Author = {J{\'a}rai, A.~A.}, Eprint = {1401.0354}, Journal = {ArXiv e-prints}, Keywords = {Mathematics - Probability, Mathematical Physics, 60K35 (primary), 82C22 (secondary)}, Month = jan, Primaryclass = {math.PR}, Title = {{Sandpile models}}, Year = 2014} @article{Jarai:2015, Abstract = {The discrete height abelian sandpile model was introduced by Bak, Tang, Wiesenfeld and Dhar as an example for the concept of self-organized criticality. When the model is modified to allow grains to disappear on each toppling, it is called bulk-dissipative. We provide a detailed study of a continuous height version of the abelian sandpile model, called the abelian avalanche model, which allows an arbitrarily small amount of dissipation to take place on every toppling. We prove that for non-zero dissipation, the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure. We show that in the whole non-zero dissipation regime, the model is not critical, i.e., spatial covariances of local observables decay exponentially. We then study the zero dissipation limit and prove that the self-organized critical model is recovered, both for the stationary measure and for the dynamics. We obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality. We rigorously establish the mean-field value of this exponent for {\$}{\$}d > 4{\$}{\$}d>4.}, Author = {J{\'a}rai, Antal A. and Redig, Frank and Saada, Ellen}, Day = {01}, Doi = {10.1007/s10955-015-1231-z}, Issn = {1572-9613}, Journal = {Journal of Statistical Physics}, Month = {Jun}, Number = {6}, Pages = {1369--1407}, Title = {Approaching Criticality via the Zero Dissipation Limit in the Abelian Avalanche Model}, Url = {https://doi.org/10.1007/s10955-015-1231-z}, Volume = {159}, Year = {2015}, Bdsk-Url-1 = {https://doi.org/10.1007/s10955-015-1231-z}} @article{Kadell85, Author = {W.J Kadell, Kevin}, Booktitle = {Journal of Combinatorial Theory, Series A}, Month = {09}, Pages = {22-44}, Title = {Weighted inversion numbers, restricted growth functions, and standard young tableaux}, Volume = {40}, Year = {1985}} @article{KALININ:2016, Author = {Nikita Kalinin and Mikhail Shkolnikov}, Doi = {https://doi.org/10.1016/j.crma.2015.11.003}, Issn = {1631-073X}, Journal = {Comptes Rendus Mathematique}, Keywords = {Combinatorics, Mathematical physics}, Number = {2}, Pages = {125 - 130}, Title = {Tropical curves in sandpiles}, Url = {http://www.sciencedirect.com/science/article/pii/S1631073X15003088}, Volume = {354}, Year = {2016}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S1631073X15003088}, Bdsk-Url-2 = {https://doi.org/10.1016/j.crma.2015.11.003}} @article{Kasteleyn:1961, Author = {Kasteleyn, P.W.}, Booktitle = {Physica}, Month = {12}, Pages = {1209-1225}, Title = {The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice}, Volume = {27}, Year = {1961}} @article{kenyon2011, Author = {Kenyon, Richard}, Doi = {10.1214/10-AOP596}, Fjournal = {The Annals of Probability}, Journal = {Ann. Probab.}, Month = {09}, Number = {5}, Pages = {1983--2017}, Publisher = {The Institute of Mathematical Statistics}, Title = {Spanning forests and the vector bundle Laplacian}, Url = {http://dx.doi.org/10.1214/10-AOP596}, Volume = {39}, Year = {2011}, Bdsk-Url-1 = {http://dx.doi.org/10.1214/10-AOP596}} @misc{kenyon2017, Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, Author = {{Kenyon}, R.}, Date-Modified = {2019-05-21 19:31:38 +1000}, Eprint = {1702.03802}, Eprinttype = {arXiv}, Journal = {ArXiv e-prints}, Keywords = {Mathematics - Probability, 82B20, 60Cxx}, Primaryclass = {math.PR}, Title = {{Determinantal spanning forests on planar graphs}}, Year = 2017} @book{Kitaev:2011, Author = {Sergey {Kitaev}}, Doi = {10.1007/978-3-642-17333-2}, Isbn = {978-3-642-17332-5/hbk; 978-3-642-17333-2/ebook}, Language = {English}, Pages = {xxii + 494}, Publisher = {Berlin: Springer}, Title = {{Patterns in permutations and words.}}, Year = {2011}, Bdsk-Url-1 = {https://doi.org/10.1007/978-3-642-17333-2}} @article{Ktitarev:1998, Author = {V. Ktitarev, D and Priezzhev, Vyatcheslav}, Booktitle = {Phys. Rev. E}, Month = {03}, Title = {Expansion and contraction of avalanches in the two-dimensional Abelian sandpile}, Volume = {58}, Year = {1998}} @book{Knuth:1997:ACP:260999, Address = {Redwood City, CA, USA}, Author = {Knuth, Donald E.}, Isbn = {0-201-89683-4}, Publisher = {Addison Wesley Longman Publishing Co., Inc.}, Title = {The Art of Computer Programming, Volume 1 (3rd Ed.): Fundamental Algorithms}, Year = {1997}} @article{LeBorgne:2002, Author = {Le~Borgne, Yvan and Rossin, Dominique}, Doi = {https://doi.org/10.1016/S0012-365X(02)00347-3}, Issn = {0012-365X}, Journal = {Discrete Mathematics}, Note = {LaCIM 2000 Conference on Combinatorics, Computer Science and Appl ications}, Number = {3}, Pages = {775 - 790}, Title = {On the identity of the sandpile group}, Url = {http://www.sciencedirect.com/science/article/pii/S0012365X02003473}, Volume = {256}, Year = {2002}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0012365X02003473}, Bdsk-Url-2 = {https://doi.org/10.1016/S0012-365X(02)00347-3}} @article{Levi2010, Author = {Levine, Propp}, Journal = {Notices of the AMS}, Title = {What is a Sandpile?}, Year = {2010}} @article{Levine:2012, Author = {Levine, Lionel and Pegden, Wesley and K. Smart, Charles}, Booktitle = {Geometric and Functional Analysis}, Month = {08}, Title = {Apollonian Structure in the Abelian Sandpile}, Volume = {26}, Year = {2012}} @article{Levine:2013, Author = {Levine, Lionel and Pegden, Wesley and K. Smart, Charles}, Booktitle = {Annals of Mathematics}, Month = {09}, Title = {The Apollonian structure of integer superharmonic matrices}, Volume = {186}, Year = {2013}} @article{Levine:2011, Author = {Levine, Lionel}, Booktitle = {Ergodic Theory and Dynamical Systems}, Month = {05}, Pages = {891 - 910}, Title = {Parallel Chip-Firing on the Complete Graph: Devil's Staircase and Poincare Rotation Number}, Volume = {31}, Year = {2011}} @article{Levine:2008, Abstract = {The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work (Levine and Peres, Indiana Univ Math J 57(1):431--450, 2008). For the shape consisting of {\$}n={\backslash}omega{\_}d r^d{\$}sites, where $\omega$dis the volume of the unit ball in {\$}{\backslash}mathbb{\{}R{\}}^d{\$}, we show that the inradius of the set of occupied sites is at least r{\thinspace}−{\thinspace}O(logr), while the outradius is at most r{\thinspace}+{\thinspace}O(r$\alpha$) for any $\alpha${\thinspace}>{\thinspace}1{\thinspace}−{\thinspace}1/d. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total mass. For the classical abelian sandpile model in two dimensions, with n{\thinspace}={\thinspace}$\pi$r2 particles, we show that the inradius is at least {\$}r/{\backslash}sqrt{\{}3{\}}{\$}, and the outradius is at most {\$}(r+o(r))/{\backslash}sqrt{\{}2{\}}{\$}. This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions, improving on bounds of Fey and Redig.}, Author = {Levine, Lionel and Peres, Yuval}, Day = {29}, Doi = {10.1007/s11118-008-9104-6}, Issn = {1572-929X}, Journal = {Potential Analysis}, Month = {Oct}, Number = {1}, Pages = {1}, Title = {Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile}, Url = {https://doi.org/10.1007/s11118-008-9104-6}, Volume = {30}, Year = {2008}, Bdsk-Url-1 = {https://doi.org/10.1007/s11118-008-9104-6}} @article{Levine:2005, Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, Adsurl = {http://adsabs.harvard.edu/abs/2005math......3251L}, Author = {{Levine}, L. and {Peres}, Y.}, Eprint = {math/0503251}, Journal = {ArXiv Mathematics e-prints}, Keywords = {Mathematics - Probability, Mathematics - Combinatorics}, Month = mar, Title = {{Spherical Asymptotics for the Rotor-Router Model in Z\^{}d}}, Year = 2005} @article{Lifschitz19811, Abstract = {The number Xn of increasing subsequences of the n-long random permutation is studied. Asymptotics of the moments of Xn are found; In Xn is shown to grow, in probability, as an, 2 ln 2 ⩽ a ⩽ 2. }, Author = {V Lifschitz and B Pittel}, Doi = {http://dx.doi.org/10.1016/0097-3165(81)90049-2}, Issn = {0097-3165}, Journal = {Journal of Combinatorial Theory, Series A}, Number = {1}, Pages = {1 - 20}, Title = {The number of increasing subsequences of the random permutation}, Url = {http://www.sciencedirect.com/science/article/pii/0097316581900492}, Volume = {31}, Year = {1981}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/0097316581900492}, Bdsk-Url-2 = {http://dx.doi.org/10.1016/0097-3165(81)90049-2}} @article{Lopez:1997, Abstract = {It is shown that the generating function of critical configurations of a version of a chip firing game on a graphG is an evaluation of the Tutte polynomial ofG, thus proving a conjecture of Biggs [3].}, Author = {L{\'o}pez, Criel Merino}, Date-Modified = {2019-05-21 19:32:16 +1000}, Doi = {10.1007/BF02558479}, Issn = {0219-3094}, Journal = {Ann. Combin.}, Number = {1}, Pages = {253--259}, Title = {Chip firing and the Tutte polynomial}, Volume = {1}, Year = {1997}, Bdsk-Url-1 = {https://doi.org/10.1007/BF02558479}} @misc{MacMahon15, Author = {P.A. MacMahon}, Publisher = {Cambridge Univ. Press}, Title = {Combinatory Analysis}, Year = {1915}} @article{Maes2004, Abstract = {We construct the thermodynamic limit of the stationary measures of the Bak-Tang-Wiesenfeld sandpile model with a dissipative toppling matrix (sand grains may disappear at each toppling). We prove uniqueness and mixing properties of this measure and we obtain an infinite volume ergodic Markov process leaving it invariant. We show how to extend the Dhar formalism of the `abelian group of toppling operators' to infinite volume in order to obtain a compact abelian group with a unique Haar measure representing the uniform distribution over the recurrent configurations that create finite avalanches}, Author = {Maes, C. and Redig, F. and Saada, E.}, Day = {01}, Doi = {10.1007/s00220-003-1000-8}, Issn = {1432-0916}, Journal = {Communications in Mathematical Physics}, Month = {Jan}, Number = {2}, Pages = {395--417}, Title = {The Infinite Volume Limit of Dissipative Abelian Sandpiles}, Url = {https://doi.org/10.1007/s00220-003-1000-8}, Volume = {244}, Year = {2004}, Bdsk-Url-1 = {https://doi.org/10.1007/s00220-003-1000-8}} @article{Majumdar1992129, Abstract = {We establish an equivalence between the undirected Abelian sandpile model and the q->0 limit of the q-state Potts model. The equivalence is valid for arbitrary finite graphs. Two-dimensional Abelian sandpile models, thus, correspond to a conformal field theory with central charge c = −2. The equivalence also gives a Monte Carlo algorithm to generate random spanning trees. We study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the Abelian sandpile model. The average number of sites burnt upto time t varies at ta. In two dimensions our numerically determined value of a agrees with the theoretical prediction a = 85. We relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes. }, Author = {S.N. Majumdar and Deepak Dhar}, Date-Modified = {2019-05-21 19:33:44 +1000}, Doi = {10.1016/0378-4371(92)90447-X}, Issn = {0378-4371}, Journal = {Physica A: Stat. Mech. Appl.}, Number = {1--4}, Pages = {129 - 145}, Title = {Equivalence between the Abelian sandpile model and the $q\to 0$ limit of the Potts model}, Volume = {185}, Year = {1992}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/037843719290447X}, Bdsk-Url-2 = {http://dx.doi.org/10.1016/0378-4371(92)90447-X}} @article{OSTOJIC2003187, Author = {Srdjan Ostojic}, Doi = {https://doi.org/10.1016/S0378-4371(02)01426-7}, Issn = {0378-4371}, Journal = {Physica A: Statistical Mechanics and its Applications}, Keywords = {Pattern formation, BTW model, Cellular automata}, Note = {STATPHYS - Kolkata IV}, Number = {1}, Pages = {187 - 199}, Title = {Patterns formed by addition of grains to only one site of an abelian sandpile}, Url = {http://www.sciencedirect.com/science/article/pii/S0378437102014267}, Volume = {318}, Year = {2003}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0378437102014267}, Bdsk-Url-2 = {https://doi.org/10.1016/S0378-4371(02)01426-7}} @phdthesis{Paoletti2012, Author = {Paoletti}, School = {Universit{\`a} di Pisa}, Title = {Deterministic Abelian Sandpile Models and Patterns}, Year = 2012} @article{Pegden:2011, Author = {Pegden, Wesley and K. Smart, Charles}, Booktitle = {Duke Mathematical Journal}, Month = {04}, Title = {Convergence of the Abelian sandpile}, Volume = {162}, Year = {2011}} @article{Pegden:2017, Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, Adsurl = {http://adsabs.harvard.edu/abs/2017arXiv170809432P}, Archiveprefix = {arXiv}, Author = {{Pegden}, W. and {Smart}, C.~K}, Eprint = {1708.09432}, Journal = {ArXiv e-prints}, Keywords = {Mathematics - Analysis of PDEs, Mathematics - Number Theory, Mathematics - Probability}, Month = aug, Primaryclass = {math.AP}, Title = {{Stability of patterns in the Abelian sandpile}}, Year = 2017} @article{Redig:2018, Author = {Redig,Frank and Ruszel,Wioletta M. and Saada,Ellen}, Doi = {10.1063/1.5022128}, Eprint = {https://doi.org/10.1063/1.5022128}, Journal = {Journal of Mathematical Physics}, Number = {6}, Pages = {063302}, Title = {Non-criticality criteria for Abelian sandpile models with sources and sinks}, Url = {https://doi.org/10.1063/1.5022128}, Volume = {59}, Year = {2018}, Bdsk-Url-1 = {https://doi.org/10.1063/1.5022128}} @article{Ruelle:2013, Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, Adsurl = {http://adsabs.harvard.edu/abs/2013JPhA...46W4014R}, Archiveprefix = {arXiv}, Author = {{Ruelle}, P.}, Doi = {10.1088/1751-8113/46/49/494014}, Eid = {494014}, Eprint = {1303.4310}, Journal = {Journal of Physics A Mathematical General}, Month = dec, Pages = {494014}, Primaryclass = {hep-th}, Title = {{Logarithmic conformal invariance in the Abelian sandpile model}}, Volume = 46, Year = 2013, Bdsk-Url-1 = {https://doi.org/10.1088/1751-8113/46/49/494014}} @article{Shapiro:1991, Author = {Shapiro, Louis and Stephens, A. B.}, Doi = {10.1137/0404025}, Fjournal = {SIAM Journal on Discrete Mathematics}, Issn = {0895-4801}, Journal = {SIAM J. Discrete Math.}, Mrclass = {05B20 (82C43)}, Mrnumber = {1093199}, Number = {2}, Pages = {275--280}, Title = {Bootstrap percolation, the {S}chr\"{o}der numbers, and the {$N$}-kings problem}, Url = {https://doi.org/10.1137/0404025}, Volume = {4}, Year = {1991}, Bdsk-Url-1 = {https://doi.org/10.1137/0404025}} @misc{Stanley2006, Author = {Richard P. Stanley}, Title = {Increasing and Decreasing Subsequences and Their Variants}, Year = {2006}} @article{tutte_1954, Author = {Tutte, W. T.}, Date-Modified = {2019-05-21 19:34:18 +1000}, Doi = {10.4153/CJM-1954-010-9}, Journal = {Canad. J. Math.}, Pages = {80--91}, Publisher = {Cambridge University Press}, Title = {A Contribution to the Theory of Chromatic Polynomials}, Volume = {6}, Year = {1954}, Bdsk-Url-1 = {https://doi.org/10.4153/CJM-1954-010-9}}