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arXiv · 1210.8213

Unlacing the lace expansion: a survey to hypercube percolation

Remco van der Hofstad·Asaf Nachmias
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Abstract

The purpose of this note is twofold. First, we survey the study of the percolation phase transition on the Hamming hypercube {0,1}^m obtained in the series of papers [9,10,11,24]. Secondly, we explain how this study can be performed without the use of the so-called "lace-expansion" technique. To that aim, we provide a novel simple proof that the triangle condition holds at the critical probability. We hope that some of these techniques will be useful to obtain non-perturbative proofs in the analogous, yet much more difficult study on high-dimensional tori.

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Remco van der Hofstad, Asaf Nachmias. 2012-10-31. Unlacing the lace expansion: a survey to hypercube percolation. https://arxiv.org/abs/1210.8213

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