From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models

Amine Asselah; Alexandre Gaudilliere

hal-00517593, version 2

From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models

Amine Asselah () ¹, Alexandre Gaudilliere () ²

(14/09/2010)

Résumé : We consider a cluster growth model on the d-dimensional lattice, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. When dimension is 2 or more, we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension d larger than or equal to 2. In so doing, we introduce a closely related cluster growth model, that we call the flashing process, whose fluctuations are controlled easily and accurately. This process is coupled to internal DLA to yield the desired bound. Part of our proof adapts the approach of Lawler, Bramson and Griffeath, on another space scale, and uses a sharp estimate (written by Blachère in our Appendix) on the expected time spent by a random walk inside an annulus.

1 : Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA)
Université Paris-Est Marne-la-Vallée (UPEMLV) – Université Paris-Est Créteil Val-de-Marne (UPEC) – CNRS : UMR8050 – Fédération de Recherche Bézout
2 : Laboratoire d'Analyse, Topologie, Probabilités (LATP)
CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III

hal-00517593, version 2
http://hal.archives-ouvertes.fr/hal-00517593
oai:hal.archives-ouvertes.fr:hal-00517593
Contributeur : Alexandre Gaudilliere <>
Soumis le : Dimanche 4 Décembre 2011, 23:37:17
Dernière modification le : Lundi 5 Décembre 2011, 10:08:34

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arXiv : 1009.2838

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%% hal-00517593, version 2
%% http://hal.archives-ouvertes.fr/hal-00517593
@unpublished{asselah:hal-00517593,
    hal_id = {hal-00517593},
    url = {http://hal.archives-ouvertes.fr/hal-00517593},
    title = {{From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models}},
    author = {Asselah, Amine and Gaudilliere, Alexandre},
    abstract = {{We consider a cluster growth model on the d-dimensional lattice, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. When dimension is 2 or more, we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension d larger than or equal to 2. In so doing, we introduce a closely related cluster growth model, that we call the flashing process, whose fluctuations are controlled easily and accurately. This process is coupled to internal DLA to yield the desired bound. Part of our proof adapts the approach of Lawler, Bramson and Griffeath, on another space scale, and uses a sharp estimate (written by Blach{\`e}re in our Appendix) on the expected time spent by a random walk inside an annulus.}},
    keywords = {internal diffusion limited aggregation, cluster growth, random walk, shape theorem, logarithmic fluctuations, subdiffusive fluctuations.},
    language = {Anglais},
    affiliation = {Laboratoire d'Analyse et de Math{\'e}matiques Appliqu{\'e}es - LAMA , Laboratoire d'Analyse, Topologie, Probabilit{\'e}s - LATP},
    note = {40 pages, 1 figure. Supported by: GDRE 224 GREFI-MEFI, the French Ministry of Education through the ANR BLAN07-2184264 grant, by the European Research Council through the "Advanced Grant" PTRELSS 228032. },
    year = {2010},
    month = Sep,
    pdf = {http://hal.archives-ouvertes.fr/hal-00517593/PDF/dla-rev.pdf},
}