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 Probability Surveys > Vol. 10 (2013) open journal systems 


Planar percolation with a glimpse of Schramm–Loewner evolution

Vincent Beffara, UMPA - ENS Lyon - CNRS
Hugo Duminil-Copin, Université de Genève


Abstract
In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievements is the proof of the Cardy--Smirnov formula. This theorem, together with the introduction of Schramm--Loewner Evolution and techniques developed over the years in percolation, allow precise descriptions of the critical and near-critical regimes of the model. This survey aims to describe the different steps leading to the proof that the infinite-cluster density \(\theta(p)\) for site percolation on the triangular lattice behaves like \((p-p_c)^{5/36+o(1)}\) as \(p\searrow p_c=1/2\).

Keywords: Site percolation, critical phenomenon, conformal invariance.

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Beffara, Vincent, Duminil-Copin, Hugo, Planar percolation with a glimpse of Schramm–Loewner evolution, Probability Surveys, 10, (2013), 1-50 (electronic). DOI: 10.1214/11-PS186.

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