2024/10/07 by David A. Croydon, Croydon, David A., Umberto De Ambroggio +1 · 1 citation
Mathematics · Computer Science · Engineering · #Stochastic processes and statistical mechanics #Optimization and Search Problems #Traffic control and management
paper · pdf · doi:10.48550/arxiv.2410.04882
In this article, we consider the number of collisions of three independent simple random walks on a subgraph of the two-dimensional square lattice obtained by removing all horizontal edges with vertical coordinate not equal to 0 and then, for n∈ ℤ, restricting the vertical segment of the graph located at horizontal coordinate n to the interval \0,1,…,logα(|n|\vee 1)\. Specifically, we show the following phase transition: when α≤ 1, the three random walks collide infinitely many times almost-surely, whereas when α>1, they collide only finitely many times almost-surely. This is a variation of a result of Barlow, Peres and Sousi, who showed a similar phase transition for two random walks when the vertical segments are truncated at height |n|α.