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Stabilization of DLA in a wedge. (English) Zbl 1441.60087
Summary: We consider diffusion limited aggregation (DLA) in a two-dimensional wedge. We prove that if the angle of the wedge is smaller than \(\pi/4\), there is some \(a>2\) such that almost surely, for all \(R\) large enough, after time \(R^a\) all new particles attached to the DLA will be at distance larger than \(R\) from the origin. Furthermore, we provide estimates on the size of \(R\) under which this holds. This means that DLA stabilizes in growing balls, thus allowing a definition of the infinite DLA in a wedge via a finite time process.
MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K40 Other physical applications of random processes
60G50 Sums of independent random variables; random walks
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