Planar percolation with a glimpse of Schramm-Loewner evolution. (English) Zbl 1283.60118

Summary: 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 the Schramm-Loewner evolution and techniques developed over the years in percolation, allows 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 \((p-p_{c})^{5/36+o(1)}= 1/2\) as \(p\searrow p_{c}\).


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
Full Text: DOI arXiv