Percolation and $O(1)$ loop model

We present an “ultimate” proof of Cardy’s formula for the critical percolation on the hexagonal lattice [23], showing the existence of the universal and conformally invariant scaling limit of crossing probabilities. The new approach is more conceptual, less technically demanding, and is amenable to generalizations.

[1]  M. Khristoforov.  Low dimensional defects in percolation model , 2018 .

[2]  Wendelin Werner,et al.  Conformal invariance of planar loop-erased random walks and uniform spanning trees , 2001 .

[3]  Stanislav Smirnov.  Critical percolation in the plane , 2009 .

[4]  H. Duminil-Copin,et al.  Convergence of Ising interfaces to Schramm's SLE curves , 2013, 1312.0533.

[5]  C. Pommerenke.  Boundary Behaviour of Conformal Maps , 1992 .

[6]  Oded Schramm.  A Percolation Formula , 2001 .

[7]  S. Smirnov,et al.  Universality in the 2D Ising model and conformal invariance of fermionic observables , 2009, 0910.2045.

[8]  H. Poincaré,et al.  Percolation ? , 1982 .

[9]  CRITICAL PERCOLATION ON MESOSCOPIC TRIANGULATIONS , 2013, 1312.7159.

[10]  Path Crossing Exponents and the External Perimeter in 2D Percolation , 1999, cond-mat/9901018.

[11]  P. Kleban,et al.  Multiple-SLE κ connectivity weights for rectangles, hexagons, and octagons , 2015, Journal of Physics A: Mathematical and Theoretical.

[12]  Yvan Saint-Aubin,et al.  Conformal invariance in two-dimensional percolation , 1994 .

[13]  J. Hammersley,et al.  Percolation processes , 1957, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  S. Smirnov.  Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits , 2001 .

[15]  V. Beffara.  Cardy's formula on the triangular lattice, the easy way , 2007 .

[16]  Clément Hongler,et al.  The energy density in the planar Ising model , 2010, 1008.2645.

[17]  John Cardy.  Critical percolation in finite geometries , 1992 .

[18]  D. Welsh,et al.  Percolation probabilities on the square lattice , 1978 .

[19]  S. Smirnov.  Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model , 2007, 0708.0039.

[20]  H. Kesten.  Percolation theory for mathematicians , 1982 .

[21]  L. Russo.  A note on percolation , 1978 .

[22]  Exploration processes and SLE$_6$ , 2014, 1409.6834.