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arXiv · 1306.3105

A lower bound on the two-arms exponent for critical percolation on the lattice

Raphaël Cerf
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Abstract

We consider the standard site percolation model on the $d$-dimensional lattice. A direct consequence of the proof of the uniqueness of the infinite cluster of Aizenman, Kesten and Newman [Comm. Math. Phys. 111 (1987) 505-531] is that the two-arms exponent is larger than or equal to $1/2$. We improve slightly this lower bound in any dimension $d\geq2$. Next, starting only with the hypothesis that $θ(p)>0$, without using the slab technology, we derive a quantitative estimate establishing long-range order in a finite box.

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Raphaël Cerf. 2015-10-29. A lower bound on the two-arms exponent for critical percolation on the lattice. https://doi.org/10.1214/14-aop940

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