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We start with a graph with a subset of vertices called the sticky set. A particle released from the origin performs nearest neighbour random walk on the graph until it comes to a nearest neighbour of the sticky set. This location becomes a new member the sticky set. In the next step, a new random walk starts from the origin and the process repeats until the origin becomes a part of the sticky set itself. We are interested in the total number $\xi$ of random walks to be released so that the origin itself becomes sticky.
We show that this model covers the \textit{OK Corral model} as well as the \textit{erosion model}. We obtain distributions and bounds for $\xi$ in cases where the graph is star graph, regular tree, and a $2-$dimensional integer lattice. For the regular binary tree, we connect the border aggregation model to the \textit{digital search tree}.
Levine and Peres (2007) observed that the border aggregation model on $d-$dimensional lattice can be considered as an "inversion" of the \textit{classical diffusion-limited-aggregation model}. |