Geodesic
Logarithmic fluctuations for internal DLA
Jerison, David1 ; Levine, Lionel1 ; Sheffield, Scott1
1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Journal of the American Mathematical Society, Tome 25 (2012) no. 1, pp. 271-301

Voir la notice de l'article provenant de la source American Mathematical Society

Let each of $n$ particles starting at the origin in $\mathbb Z^2$ perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set $A(n)$ of $n$ occupied sites is (with high probability) close to a disk $\mathbf {B}_r$ of radius $r=\sqrt {n/\pi }$. We show that the discrepancy between $A(n)$ and the disk is at most logarithmic in the radius: i.e., there is an absolute constant $C$ such that with probability $1$, \[ \mathbf {B}_{r - C\log r} \subset A(\pi r^2) \subset \mathbf {B}_{r+ C\log r} \quad \mbox { for all sufficiently large $r$}. \]
DOI : 10.1090/S0894-0347-2011-00716-9

Jerison, David 1 ; Levine, Lionel 1 ; Sheffield, Scott 1

1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 
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Jerison, David; Levine, Lionel; Sheffield, Scott. Logarithmic fluctuations for internal DLA. Journal of the American Mathematical Society, Tome 25 (2012) no. 1, pp. 271-301. doi: 10.1090/S0894-0347-2011-00716-9

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