Border aggregation model. (English) Zbl 1410.60098
Summary: Start with a graph with a subset of vertices called the border. A particle released from the origin performs a random walk on the graph until it comes to the immediate neighbourhood of the border, at which point it joins this subset thus increasing the border by one point. Then a new particle is released from the origin and the process repeats until the origin becomes a part of the border itself. We are interested in the total number \(\xi\) of particles to be released by this final moment.
We show that this model covers the OK Corral model as well as the erosion model, and obtain distributions and bounds for \(\xi\) in cases where the graph is star graph, regular tree and a \(d\)-dimensional lattice.
We show that this model covers the OK Corral model as well as the erosion model, and obtain distributions and bounds for \(\xi\) in cases where the graph is star graph, regular tree and a \(d\)-dimensional lattice.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 82B24 | Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics |
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