Internal DLA process converges to a disk with at most logarithmic discrepancy

Probability and Statistics 1 Minute

You need to know: Integer lattice {\mathbb Z}^2, origin 0, basic probability theory, simple random walk on {\mathbb Z}^2, notation \left \lfloor{t}\right \rfloor be the largest integer not exceeding t.

Background: The internal diffusion limited aggregation process (internal DLA process for short) is defined as follows. For each integer time n\geq 1, construct a random set A(n) \subset {\mathbb Z}^2 such that (i) A(1) = \{0\}, and (ii) for each n\geq 1, let A(n+1) be A(n) plus the first point at which a simple random walk from the origin hits {\mathbb Z}^2 \setminus A(n). For any real t\geq 1, let A(t)=A(\left \lfloor{t}\right \rfloor). For r>0, let B(r)=\{z=(z_1,z_2) \in {\mathbb Z}^2\,|\,z_1^2+z_2^2 < r^2\}.

The Theorem: On 12th October 2010, David Jerison, Lionel Levine, and Scott Sheffield submitted to arxiv a paper in which they proved the existence of an absolute constant C such that with probability 1, B(r-C\log r) \subset A(\pi r^2) \subset B(r+C\log r) for all sufficiently large r.

Short context: Internal DLA process was proposed by Meakin and Deutch in 1986 as a model of industrial chemical processes. They found numerically that, for large n, the process becomes close to a disk with at most logarithmic fluctuations. In 1992, Lawler, Bramson and Griffeath proved that the asymptotic shape of the domain A(n) is indeed a disk. The Theorem confirms that the fluctuations are indeed at most logarithmic.

Links: Free arxiv version of the paper is here, journal version is here.

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Published by Bogdan Grechuk

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