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September 2015 A lower bound on the two-arms exponent for critical percolation on the lattice
Raphaël Cerf
Ann. Probab. 43(5): 2458-2480 (September 2015). DOI: 10.1214/14-AOP940

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 $\theta(p)>0$, without using the slab technology, we derive a quantitative estimate establishing long-range order in a finite box.

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Raphaël Cerf. "A lower bound on the two-arms exponent for critical percolation on the lattice." Ann. Probab. 43 (5) 2458 - 2480, September 2015. https://doi.org/10.1214/14-AOP940

Information

Received: 1 July 2013; Revised: 1 May 2014; Published: September 2015
First available in Project Euclid: 9 September 2015

zbMATH: 1356.60163
MathSciNet: MR3395466
Digital Object Identifier: 10.1214/14-AOP940

Subjects:
Primary: 60K35
Secondary: 82B43

Keywords: arms exponent , Critical percolation

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 5 • September 2015
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