Abstract
We prove a law of large numbers for the right boundary in the model of internal DLA generated by cookie random walks in dimension one. The proof is based on stochastic recursions techniques.
Highlights
We prove a law of large numbers for the right boundary in the model of internal DLA generated by cookie random walks in dimension one
The model of Internal DLA on Zd is a basic example of cluster growth model, which has the Euclidean ball as limiting shape, and for which we have very good control on the size of the fluctuations around this limiting shape
On a mathematical level at least, by Diaconis and Fulton [9] in dimension one. In this case the cluster at any time is an interval, and the analysis of its extremities can be transposed in terms of Polya’s urn, for which we know that the law of large numbers and the central limit theorem hold
Summary
The model of Internal DLA (internal diffusion limited aggregation) on Zd is a basic example of cluster growth model, which has the Euclidean ball as limiting shape, and for which we have very good control (presumably sharp) on the size of the fluctuations around this limiting shape It was introduced, on a mathematical level at least, by Diaconis and Fulton [9] in dimension one. For instance Blachère and Brofferio [6] consider graphs with exponential growth and Shellef [22] looked at the case of a random graph, namely a supercritical cluster of percolation In another direction it is interesting to see how the analysis can be transposed when we change the simple random walk by some other random process.
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