Project Euclid

References

• [1] Per Bak, Chao Tang, and Kurt Wiesenfeld, Self-organized criticality, Phys. Rev. A (3) 38 (1988), no. 1, 364–374.
• [2] B. Bond and L. Levine, Abelian networks: foundations and examples, ArXiv e-prints (2013).
• [3] M. Casartelli, L. Dall’Asta, A. Vezzani, and P. Vivo, Dynamical invariants in the deterministic fixed-energy sandpile, European Physical Journal B 52 (2006), 91–105.
• [4] L. Dall’Asta, Exact Solution of the One-Dimensional Deterministic Fixed-Energy Sandpile, Physical Review Letters 96 (2006), no. 5, 058003.
• [5] Deepak Dhar, Self-organized critical state of sandpile automaton models, Phys. Rev. Lett. 64 (1990), no. 14, 1613–1616.
• [6] P. Diaconis and W. Fulton, A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Sem. Mat. Univ. Politec. Torino 49 (1991), no. 1, 95–119 (1993), Commutative algebra and algebraic geometry, II (Italian) (Turin, 1990).
• [7] Ronald Dickman, Leonardo T. Rolla, and Vladas Sidoravicius, Activated random walkers: facts, conjectures and challenges, J. Stat. Phys. 138 (2010), nos 1–3, 126–142.
• [8] Kimmo Eriksson, Chip-firing games on mutating graphs, SIAM J. Discrete Math. 9 (1996), no. 1, 118–128.
  Mathematical Reviews (MathSciNet): MR1375419
  Digital Object Identifier: doi:10.1137/S0895480192240287
  
• [9] A. Fey, L. Levine, and D. B. Wilson, Driving Sandpiles to Criticality and Beyond, Physical Review Letters 104 (2010), no. 14, 145703.
• [10] A. Fey and R. Meester, Critical densities in sandpile models with quenched or annealed disorder, ArXiv e-prints (2012).
• [11] Marcelo Hilário, Frank Den Hollander, Vladas Sidoravicius, Renato Soares dos Santos, and Augusto Teixeira, Random walk on random walks., Electron. J. Probab. 20 (2015), 35 (English).
• [12] Lionel Levine, Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle, Communications in Mathematical Physics 335 (2015), no. 2, 1003–1017.
• [13] Lionel Levine and Yuval Peres, Scaling limits for internal aggregation models with multiple sources, J. Anal. Math. 111 (2010), 151–219.
• [14] Thomas M. Liggett, Interacting particle systems, Classics in Mathematics, Springer-Verlag, Berlin, 2005, Reprint of the 1985 original.
• [15] S S Manna, Two-state model of self-organized criticality, Journal of Physics A: Mathematical and General 24 (1991), no. 7, L363.
• [16] Colin McDiarmid, Concentration, Probabilistic methods for algorithmic discrete mathematics, Algorithms Combin., vol. 16, Springer, Berlin, 1998, pp. 195–248.
• [17] R. Meester and C. Quant, Connections between ‘self-organised’ and ‘classical’ criticality, Markov Process. Related Fields 11 (2005), no. 2, 355–370.
• [18] M. V. Men$'$shikov, Coincidence of critical points in percolation problems, Dokl. Akad. Nauk SSSR 288 (1986), no. 6, 1308–1311.
• [19] Jason Miller and Yuval Peres, Uniformity of the uncovered set of random walk and cutoff for lamplighter chains, Ann. Probab. 40 (2012), no. 2, 535–577.
• [20] M. A. Muñoz, R. Dickman, R. Pastor-Satorras, A. Vespignani, and S. Zapperi, Sandpiles and absorbing-state phase transitions: Recent results and open problems, American Institute of Physics Conference Series, American Institute of Physics Conference Series, vol. 574, June 2001, pp. 102–110.
• [21] Yuval Peres, Alistair Sinclair, Perla Sousi, and Alexandre Stauffer, Mobile geometric graphs: detection, coverage and percolation, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (Philadelphia, PA), SIAM, 2011, pp. 412–428.
• [22] Su. Poghosyan, V. Poghosyan, V. Priezzhev, and P. Ruelle, Numerical study of the correspondence between the dissipative and fixed-energy abelian sandpile models, Phys. Rev. E 84 (2011), 066119.
• [23] Serguei Popov and Augusto Teixeira, Soft local times and decoupling of random interlacements, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 10, 2545–2593 (English).
• [24] James Gary Propp and David Bruce Wilson, How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph, J. Algorithms 27 (1998), no. 2, 170–217, 7th Annual ACM-SIAM Symposium on Discrete Algorithms (Atlanta, GA, 1996).
• [25] Frank Redig, Mathematical aspects of the abelian sandpile model, Mathematical statistical physics, Elsevier B. V., Amsterdam, 2006, pp. 657–729.
• [26] Sidney I. Resnick, Extreme values, regular variation and point processes, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2008, Reprint of the 1987 original.
• [27] L. T. Rolla, Generalized Hammersley Process and Phase Transition for Activated Random Walk Models, Ph.D. thesis, 2008.
• [28] L. T. Rolla and V. Sidoravicius, Absorbing-state phase transition for driven-dissipative stochastic dynamics on Z, Inventiones Mathematicae 188 (2012), 127–150.
• [29] Yu. A. Rozanov, Markov random fields, Applications of Mathematics, Springer-Verlag, New York, 1982, Translated from Russian by Constance M. Elson.
• [30] Eric Shellef, Nonfixation for activated random walks, ALEA Lat. Am. J. Probab. Math. Stat. 7 (2010), 137–149.
• [31] Didier Sornette, Critical phenomena in natural sciences, second ed., Springer Series in Synergetics, Springer-Verlag, Berlin, 2006, Chaos, fractals, selforganization and disorder: concepts and tools.
• [32] A. Stauffer and L. Taggi, Critical density of activated random walks on $\mathbb{Z} ^d$ and general graphs, ArXiv e-prints (2015).