The unreasonable effectiveness of martingales
نویسنده
چکیده
Proof. Consider p-bond percolation on an an infinite dregular tree T rooted at ρ and write CT (ρ) for the component containing ρ. Write CG(v) for the component of v in Gp. It is clear that we can couple such that |CG(v)| ≤ |CT (ρ)|. Recall the exploration process on T : At time t = 0 only ρ = v1 is active, other vertices neutral. At time t ≥ 0 if the number of active vertices Mt > 0, then set the first active vt explored, and activate all its neutral neighbors, so
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