Advanced Topics in Probability (Conformal Methods in 2D Statistical ...

Advanced Topics in Probability (Conformal Methods in 2D Statistical Mechanics)

References [1] W. Werner, Lectures on two-dimensional critical percolation, in Statistical Mechanics, IAS / Park City Mathematics series vol. 16 (2009) (available at arXiv:0710.0856). [2] G.F. Lawler, Schramm-Loewner Evolution (SLE), in Statistical Mechanics, IAS / Park City Mathematics series vol. 16 (2009) (available at arXiv:0712.3256). [3] W. Werner, Percolation et modèle d’Ising, Cours spécialisés, SMF (2009) (in French). [4] G.R. Grimmett, Percolation, 2nd edition, Springer (1999). [5] G.R. Grimmett, The random-cluster model, Springer (2006). [6] H. Kesten, Percolation theory for mathematicians, Birkhäuser (1982). [7] G.F. Lawler, Conformally invariant processes in the plane, AMS (2005). [8] W. Werner, Random planar curves and Schramm-Loewner Evolutions, in 2002 Saint-Flour summer school, Lecture Notes in Mathematics 1840, Springer (2004).

. . . and some of the original papers [1] M. Aizenman, A. Burchard, Hölder regularity and dimension bounds for random curves, Duke Math. J. 99, 419-453 (1999). [2] V. Beffara, Hausdorff dimensions for SLE6 , Ann. Probab. 32, 2606-2629 (2004). [3] V. Beffara, The dimension of the SLE curves, Ann. Probab. 36, 1421-1452 (2008). [4] V. Beffara, H. Duminil-Copin, The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1, preprint arXiv:1006.5073 (2010). [5] J. Van den Berg, H. Kesten, Inequalities with applications to percolation and reliability, J. Appl. Probab. 22, 556-569 (1985). 1

[6] F. Camia, C.M. Newman, Two-dimensional critical percolation: the full scaling limit, Comm. Math. Phys. 268, 1-38 (2006). [7] F. Camia, C.M. Newman, Critical percolation exploration path and SLE6 : a proof of convergence, Probab. Th. Rel. Fields 139, 473-519 (2007). [8] D. Chelkak, S. Smirnov, Discrete complex analysis on isoradial graphs, preprint arXiv:0810.2188 (2008). [9] D. Chelkak, S. Smirnov, Universality in the 2D Ising model and conformal invariance of fermionic observables, preprint arXiv:0910.2045 (2009). [10] H. Duminil-Copin, p S.√Smirnov, The connective constant of the honeycomb lattice equals 2 + 2, preprint arXiv:1007.0575 (2010). [11] H. Kesten, The critical probability of bond percolation on the square lattice equals 1/2, Comm. Math. Phys. 74, 41-59 (1980). [12] H. Kesten, Scaling relations for 2D-percolation, Comm. Math. Phys. 109, 109-156 (1987). [13] G.F. Lawler, O. Schramm, W. Werner, Values of Brownian intersection exponents I: Half-plane exponents, Acta Mathematica 187, 237-273 (2001). [14] G.F. Lawler, O. Schramm, W. Werner, Values of Brownian intersection exponents II: Plane exponents, Acta Mathematica 187, 275-308 (2001). [15] G.F. Lawler, O. Schramm, W. Werner, One-arm exponent for critical 2D percolation, Elec. J. Probab. 7, paper no.2 (2002). [16] G.F. Lawler, O. Schramm, W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. 32, 939-995 (2004). [17] S. Rohde, O. Schramm, Basic properties of SLE, Ann. Math. 161, 883-924 (2005). [18] O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118, 221-288 (2000). [19] S. Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333, 239-244 (2001). [20] S. Smirnov, Towards conformal invariance of 2D lattice models, Proc. ICM 2006, vol. 2, 1421-1451 (2007). [21] S. Smirnov, Conformal invariance in FK models. I. Holomorphic fermions in the Ising model, Ann. Math., to appear. [22] S. Smirnov, W. Werner, Critical exponents for two-dimensional percolation, Math. Res. Lett. 8, 729-744 (2001).

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