Is Critical 2D Percolation Universal?
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.
KeywordsPercolation Conformal invariance Complex structure
Mathematics Subject Classification (2000)82B43 32G15 82B27
Unable to display preview. Download preview PDF.
- Vincent Beffara, Quantitative estimates for the incipient infinite cluster of 2D percolation, in preparation.Google Scholar
- -Cardy’s formula on the triangular lattice, the easy way, Universality and Renormalization (Hia Binder and Dirk Kreimer eds.) Fields Institute Communications, vol. 50, The Fields Institute, 2007, pp. 39–45.Google Scholar
- Béla Bollobás and Oliver Riordan, Percolation, Cambridge University Press, 2006.Google Scholar
- Harry Kesten, Percolation theory for mathematicians, Progress in Probability and Statistics, vol. 2, Birkhäuser Boston, Mass., 1982.Google Scholar
- -Critical percolation in the plane. I. Conformal invariance and Cardy’s formula. II. Continuum scaling limit, http://www.math.kth.se/stas/papers/percol.ps, 2001.Google Scholar
- Kenneth Stephenson, Introduction to circle packing, Cambridge University Press, 2005.Google Scholar