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.
"Border aggregation model." Ann. Appl. Probab. 28 (3) 1604 - 1633, June 2018. https://doi.org/10.1214/17-AAP1339