## 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 It relies on an unpublished lemma of O. Schramm.

### MSC:

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

### Citations:

Zbl 0926.60004; Zbl 0586.60096
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### References:

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