×

Internal diffusion-limited aggregation with uniform starting points. (English. French summary) Zbl 1434.82062

Summary: We study internal diffusion-limited aggregation with uniform starting points on \(\mathbb{Z}^d \). In this model, each new particle starts from a vertex chosen uniformly at random on the existing aggregate. We prove that the limiting shape of the aggregate is a Euclidean ball.

MSC:

82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
60J45 Probabilistic potential theory
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] A. Asselah and A. Gaudilliere. From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models. Ann. Probab. 41 (3A) (2013) 1115-1159. · Zbl 1283.60117
[2] A. Asselah and A. Gaudilliere. Sublogarithmic fluctuations for internal DLA. Ann. Probab. 41 (3A) (2013) 1015-1059. · Zbl 1274.60286
[3] S. Blachère. Internal diffusion limited aggregation on discrete groups of polynomial growth. In Random Walks and Geometry: Proceedings of a Workshop at the Erwin Schrödinger Institute, Vienna, June 18-July 13, 2001 377. De Gruyter, Berlin, 2004. Available at degruyter.com/14448. · Zbl 1061.60111
[4] S. Blachère and S. Brofferio. Internal diffusion limited aggregation on discrete groups having exponential growth. Probab. Theory Related Fields 137 (3) (2007) 323-343. · Zbl 1106.60078
[5] O. Couronné, N. Enriquez and L. Gerin. Construction of a short path in high-dimensional first passage percolation. Electron. Commun. Probab. 16 (2011) 22-28. · Zbl 1231.60109
[6] P. Diaconis and W. Fulton. A growth model, a game, an algebra, Lagrange inversion, and characteristic classes. Rend. Semin. Mat. Univ. Politec. Torino 49 (1) (1991) 95-119. · Zbl 0776.60128
[7] H. Duminil-Copin, C. Lucas and A. Yadin. Internal diffusion limited aggregation on groups of polynomial growth. Preprint, 2013. · Zbl 1300.60065
[8] H. Duminil-Copin, C. Lucas, A. Yadin and A. Yehudayoff. Containing internal diffusion limited aggregation. Electron. Commun. Probab. 18 (2013) 50. · Zbl 1300.60065
[9] M. Eden. A two-dimensional growth process. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. IV 223-239. Univ. California Press, Berkeley, 1961. Available at projecteuclid.org/1200512888. · Zbl 0104.13801
[10] G. Fayolle and M. Krikun. Growth rate and ergodicity conditions for a class of random trees. Math. Comput. Sci. II (2002) 381-391. · Zbl 1030.05106
[11] W. Huss. Internal diffusion-limited aggregation on non-amenable graphs. Electron. Commun. Probab. 13 (2008) 272-279. · Zbl 1189.60178
[12] D. Jerison, L. Levine and S. Sheffield. Logarithmic fluctuations for internal DLA. J. Amer. Math. Soc. 25 (1) (2012) 271-301. · Zbl 1237.60037
[13] D. Jerison, L. Levine and S. Sheffield. Internal DLA in higher dimensions. Electron. J. Probab. 18 (98) (2013) 14. · Zbl 1290.60051
[14] D. Jerison, L. Levine and S. Sheffield. Internal DLA and the Gaussian free field. Duke Math. J. 163 (2) (2014) 267-308. · Zbl 1296.60113
[15] H. Kesten. Aspects of first passage percolation. In École d’été de probabilités de Saint-Flour, XIV - 1984 125-264. Springer Lecture Notes in Math. 1180, 1986. Available at http://link.springer.com/chapter/10.1007/BFb0074919. · Zbl 0602.60098
[16] G. F. Lawler. Subdiffusive fluctuations for internal diffusion limited aggregation. Ann. Probab. 23 (1) (1995) 71-86. · Zbl 0835.60086
[17] G. F. Lawler, M. Bramson and D. Griffeath. Internal diffusion limited aggregation. Ann. Probab. 20 (4) (1992) 2117-2140. · Zbl 0762.60096
[18] G. F. Lawler and V. Limic. Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge, 2010. · Zbl 1210.60002
[19] L. Levine and Y. Peres. Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile. Potential Anal. 30 (1) (2009) 1-27. · Zbl 1165.82309
[20] C. Lucas. The limiting shape for drifted internal diffusion limited aggregation is a true heat ball. Probab. Theory Related Fields 159 (1-2) (2014) 197-235. · Zbl 1296.60115
[21] D. Richardson. Random growth in a tesselation. Proc. Camb. Philos. Soc. 74 (1973) 515-528. · Zbl 0295.62094
[22] E. Shellef. Idla on the supercritical percolation cluster. Electron. J. Probab. 15 (2010) 723-740. · Zbl 1226.60136
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.