Planar Site Percolation via Tree Embeddings

Zhongyang Li
University of Connecticut

I will show that for any infinite, connected, planar graph G properly embedded into the plane with a minimal vertex degree of at least 7, the i.i.d. Bernoulli(p) site percolation on G almost surely (a.s.) has infinitely many infinite 1-clusters for any p\in (p_c^{site},1-p_c^{site}). Moreover, p_c^{site}<1/2 , so the above interval is non-empty. This confirms a conjecture by Benjamini and Schramm in 1996. The proof is based on a novel construction of embedded trees on such graphs.

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