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Dispersion processes. (English) Zbl 1405.05170

Summary: We study a synchronous process called dispersion. Initially \(M\) particles are placed at a distinguished origin vertex of a graph \(G\). At each time step, at each vertex \(v\) occupied by more than one particle at the beginning of this step, each of these particles moves to a neighbor of \(v\) chosen independently and uniformly at random. The process ends at the first step when no vertex is occupied by more than one particle. For the complete graph \(K_n\), for any constant \(\delta>1\), with high probability, if \(M\leq n/2(1-\delta)\), the dispersion process finishes in \(O(\log n)\) steps, whereas if \(M\geq n/2(1+\delta)\), the process needs \(e^{\Omega(n)}\) steps to complete, if ever. A lazy variant of the process exhibits analogous behavior but at a higher threshold, thus allowing faster dispersion of more particles. For paths, trees, grids, hypercubes, and abelian Cayley graphs of large enough size, we give bounds on the time to finish and the maximum distance traveled from the origin as a function of \(M\).

MSC:

05C81 Random walks on graphs
05C80 Random graphs (graph-theoretic aspects)
05C85 Graph algorithms (graph-theoretic aspects)
05C12 Distance in graphs
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