Percolation on a product of two trees. (English) Zbl 1243.60078

Using the standard concepts of percolation theory (see, e.g., [G. Grimmett, Percolation. 2nd ed. Berlin: Springer. (1999; Zbl 0926.60004]) the author studies the triangle condition validity. One says that a transitive graph \(G\) satisfies the triangle condition at some \(p\) if \[ \nabla_p:=\sum_{x,y\in G}\mathbb{P}_p(0\longleftrightarrow x)\mathbb{P}_p(x\longleftrightarrow y) \mathbb{P}_p(0\longleftrightarrow y)<\infty. \] M. Aizenman and C. M. Newman [J. Stat. Phys. 36, 107–143 (1984; Zbl 0586.60096] suggested this condition as a marker for “mean-field behavior”. In particular, they proved that if \(\nabla_{p_c}<\infty\) then \(\mathbb{E}_p|C(0)|\approx (p_c-p)^{-1}\) as \(p\) tends to the critical probability \(p_c\) from below. Here \(C(0)=\{x:0\longleftrightarrow x\}\) and \(|C(0)|\) is its size. In the present paper G. Kozma established that if \(T\) is a regular tree of degree greater than two, then the product graph \(T\times T\) satisfies the triangle condition at \(p_c\). The proof does not examine the degree of vertices and is not “perturbative” in any sense. It relies on an unpublished lemma of O. Schramm.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60B99 Probability theory on algebraic and topological structures
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