{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,24]],"date-time":"2026-10-24T00:52:56Z","timestamp":1792803176628,"version":"4.5.0"},"reference-count":13,"publisher":"Duke University Press","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Duke Math. J."],"published-print":{"date-parts":[[2013,3,15]]},"abstract":"<jats:p>The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice Zd, in which sites with at least 2d chips topple, distributing one chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of n chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as n\u2192\u221e. However, little has been proved about the appearance of the stable configurations. We use partial differential equation techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as n\u2192\u221e. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.<\/jats:p>","DOI":"10.1215\/00127094-2079677","type":"journal-article","created":{"date-parts":[[2013,3,15]],"date-time":"2013-03-15T10:30:29Z","timestamp":1363343429000},"page":"627-642","source":"Crossref","is-referenced-by-count":46,"title":["Convergence of the Abelian sandpile"],"prefix":"10.1215","volume":"162","author":[{"given":"Wesley","family":"Pegden","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Charles K.","family":"Smart","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"73","reference":[{"key":"1","doi-asserted-by":"publisher","unstructured":"[1] P. Bak, C. Tang, and K. Wiesenfeld, <i>Self-organized criticality<\/i>, Phys. Rev. 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Fulton, \u201cA growth model, a game, an algebra, Lagrange inversion, and characteristic classes\u201d in <i>Commutative Algebra and Algebraic Geometry, II (Turin, 1990)<\/i>, Rend. Semin. Mat. Univ. Politec. Torino <b>49<\/b>, Libreria Editrice Universitaria Levrotto &amp; Bella, Turin, 1993, 95\u2013119."},{"key":"6","doi-asserted-by":"crossref","unstructured":"[6] L. C. Evans, <i>Weak Convergence Methods for Nonlinear Partial Differential Equations<\/i>, CBMS Reg. Conf. Ser. Math. <b>74<\/b>, Amer. Math. Soc., Providence, 1990.","DOI":"10.1090\/cbms\/074"},{"key":"7","doi-asserted-by":"publisher","unstructured":"[7] A. Fey, L. Levine, and Y. Peres, <i>Growth rates and explosions in sandpiles<\/i>, J. Stat. Phys. <b>138<\/b> (2010), 143\u2013159.","DOI":"10.1007\/s10955-009-9899-6"},{"key":"8","doi-asserted-by":"crossref","unstructured":"[8] D. Gilbarg and N. S. Trudinger, <i>Elliptic Partial Differential Equations of Second Order<\/i>, Classics Math., Springer, Berlin, 2001.","DOI":"10.1007\/978-3-642-61798-0"},{"key":"9","doi-asserted-by":"publisher","unstructured":"[9] H.-J. Kuo and N. S. Trudinger, \u201cEstimates for solutions of fully nonlinear discrete schemes\u201d in <i>Trends in Partial Differential Equations of Mathematical Physics<\/i>, Progr. Nonlinear Differential Equations Appl. <b>61<\/b>, Birkh\u00e4user, Basel, 2005, 275\u2013282.","DOI":"10.1007\/3-7643-7317-2_20"},{"key":"10","doi-asserted-by":"crossref","unstructured":"[10] G. F. Lawler and V. Limic, <i>Random Walk: A Modern Introduction<\/i>, Cambridge Stud. Adv. Math. <b>123<\/b>, Cambridge Univ. Press, Cambridge, 2010.","DOI":"10.1017\/CBO9780511750854"},{"key":"11","doi-asserted-by":"publisher","unstructured":"[11] L. Levine and Y. Peres, <i>Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile<\/i>, Potential Anal. <b>30<\/b> (2009), 1\u201327.","DOI":"10.1007\/s11118-008-9104-6"},{"key":"12","unstructured":"[12] L. Levine and J. Propp, <i>What is \u2026 a sandpile?<\/i> Notices Amer. Math. Soc. <b>57<\/b> (2010), 976\u2013979."},{"key":"13","doi-asserted-by":"crossref","unstructured":"[13] S. Ostojic, <i>Patterns formed by addition of grains to only one site of an Abelian sandpile<\/i>, Phys. A <b>318<\/b> (2003), 187\u2013199.","DOI":"10.1016\/S0378-4371(02)01426-7"}],"container-title":["Duke Mathematical Journal"],"original-title":[],"link":[{"URL":"https:\/\/projecteuclid.org\/journalArticle\/Download?urlid=10.1215\/00127094-2079677","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T18:13:44Z","timestamp":1791569624000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/duke-mathematical-journal\/volume-162\/issue-4\/Convergence-of-the-Abelian-sandpile\/10.1215\/00127094-2079677.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,3,15]]},"references-count":13,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2013,3,15]]}},"URL":"https:\/\/doi.org\/10.1215\/00127094-2079677","relation":{},"ISSN":["0012-7094","1547-7398"],"issn-type":[{"value":"0012-7094","type":"print"},{"value":"1547-7398","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,3,15]]}}}