Internal Diffusion-Limited Aggregation on non-amenable graphs

Wilfried Huss (Graz University of Technology)

Abstract


The stochastic growth model Internal Diffusion Limited Aggregation was defined in 1991 by Diaconis and Fulton. Several shape results are known when the underlying state space is the d-dimensional lattice, or a discrete group with exponential growth. We prove an extension of the shape result of Blachere and Brofferio for Internal Diffusion Limited Aggregation on a wide class of Markov chains on non-amenable graphs.

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Pages: 272-279

Publication Date: May 25, 2008

DOI: 10.1214/ECP.v13-1374

References

  1. Diaconis, P.; Fulton, W. A growth model, a game, an algebra, Lagrange inversion, and characteristic classes. Commutative algebra and algebraic geometry, II (Italian) (Turin, 1990). Rend. Sem. Mat. Univ. Politec. Torino 49 (1991), no. 1, 95--119 (1993). MR1218674 (94d:60105)
  2. Lawler, Gregory F.; Bramson, Maury; Griffeath, David. Internal diffusion limited aggregation. Ann. Probab. 20 (1992), no. 4, 2117--2140. MR1188055 (94a:60105)
  3. Lawler, Gregory F. Subdiffusive fluctuations for internal diffusion limited aggregation. Ann. Probab. 23 (1995), no. 1, 71--86. MR1330761 (96c:60086)
  4. Blachère, Sébastien. Agrégation limitée par diffusion interne et temps de coupure sur les groupes discrets à croissance polynomiale. PhD thesis, L'Universitée Paul Sabatier, 2000. Math. Review number not available.
  5. Blachère, Sébastien. Internal diffusion limited aggregation on discrete groups having polynomial growth. in Random Walks and Geometry, Proceedings (Erwin Schroedinger Insitute, Vienna 2001). de Gruyter, Berlin, 2004. Math. Review number not available.
  6. Blachère, Sébastien; Brofferio, Sara. Internal diffusion limited aggregation on discrete groups having exponential growth. Probab. Theory Related Fields 137 (2007), no. 3-4, 323--343. MR2278460 (2008b:60210)
  7. Blachère, S; Haïssinsky, P; Mathieu, P. Asymptotic entropy and Green speed for random walks on countable groups. Ann. Probab. 36 (2008) n.3 1134--1152. Math. Review number not available.
  8. Woess, Wolfgang. Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp. ISBN: 0-521-55292-3 MR1743100 (2001k:60006)
  9. Dodziuk, Jozef. Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc. 284 (1984), no. 2, 787--794. MR0743744 (85m:58185)
  10. Dodziuk, J.; Kendall, W. S. Combinatorial Laplacians and isoperimetric inequality. From local times to global geometry, control and physics (Coventry, 1984/85), 68--74, Pitman Res. Notes Math. Ser., 150, Longman Sci. Tech., Harlow, 1986. MR0894523 (88h:58118)
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