Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, May 17, 2016, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Ahad N. Zehmakan

Color War on Grid

You are Given a graph $G(V,E)$ and an initial random vertex-coloring such that each vertex $v\in V$ is blue with probability $p$ and red with probability $1-p$. In each step, all vertices change their current color synchronously to the most frequent color in their neighborhood (in the case of a tie, a vertex conserves its current color). We discuss the asymptotic behavior of this (random) process which is called color war. We mostly focus on grid and torus (probabilistic cellular automata) and, as a main result, show there exits $0 < q < 1$ as a threshold such that if $p \ll q$ a grid (torus) $G(V,E)$ reaches a red generation (all vertices red) in a constant number of steps, but $p \gg q$ results in the coexistence of both colors in a configuration of period one or two in at most $2|V|$ steps. You can also assume in the case of a tie, a vertex flips a coin to choose blue or red. We name this Markov chain random color war, and present also some interesting properties regarding its behavior.

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