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Is critical 2D percolation universal? (English) Zbl 1173.82327

Sidoravicius, Vladas (ed.) et al., In and out of equilibrium 2. Papers celebrating the 10th edition of the Brazilian school of probability (EBP), Rio de Janiero, Brazil, July 30 to August 4, 2006. Basel: Birkhäuser (ISBN 978-3-7643-8785-3/hbk). Progress in Probability 60, 31-58 (2008).
Summary: The aim of these notes is to explore possible ways of extending Smirnov’s proof of Cardy’s formula for critical site-percolation on the triangular lattice to other cases (such as bond-percolation on the square lattice); the main question we address is that of the choice of the lattice embedding into the plane which gives rise to conformal invariance in the scaling limit. Even though we were not able to produce a complete proof, we believe that the ideas presented here go in the right direction.
For the entire collection see [Zbl 1141.82002].

MSC:

82B43 Percolation
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
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