%0 Book Section %1 statphys23_1048 %A Ziff, R.M. %A Scullard, C.M. %A Kleban, P. %A Simmons, J.J.H. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K ising model percolation potts statphys23 thresholds topic-2 universality %T Crossing problems and thresholds in percolation and the Potts model %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1048 %X Various crossing problems in percolation and on Fortuin-Kasteleyn clusters of the Potts and Ising models is discussed. For certain boundary conditions in the Potts case, a perfectly dual system can be made so that the crossing probability for a square system at the critical threshold is exactly 1/2, as it is for simple independent bond percolation at the threshold. Likewise, a new scaled probability parameter can be constructed to give symmetric results for the crossing, just like percolation. The crossing probability for rectangular systems is shown to agree with theory Gruzberg. For horizontal-but-not-vertical crossing, the asymptotic decay of the crossing probability is found numerically -- here, there are no theoretical results. Another advance in percolation is also presented: new lattices where the thresholds are known exactly, namely the martini lattice and its relatives work with C. Scullard and generalizations of the bow-tie lattice of Wierman. Self-dual systems of these lattices are constructed, and the duality of the crossing probability leads to the new exact thresholds. References: 1. R. M. Ziff and C. R. Scullard, J. Phys. A 39, 15083 (2006) 2. J. C. Wierman, J. Phys. A 17, 1525 (1984) 3. Ilya A. Gruzberg, Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory, arXiv:math-ph/0607046

@incollection{statphys23_1048, abstract = {Various crossing problems in percolation and on Fortuin-Kasteleyn clusters of the Potts and Ising models is discussed. For certain boundary conditions in the Potts case, a perfectly dual system can be made so that the crossing probability for a square system at the critical threshold is exactly 1/2, as it is for simple independent bond percolation at the threshold. Likewise, a new scaled probability parameter can be constructed to give symmetric results for the crossing, just like percolation. The crossing probability for rectangular systems is shown to agree with theory [Gruzberg]. For horizontal-but-not-vertical crossing, the asymptotic decay of the crossing probability is found numerically -- here, there are no theoretical results. Another advance in percolation is also presented: new lattices where the thresholds are known exactly, namely the martini lattice and its relatives [work with C. Scullard] and generalizations of the bow-tie lattice of Wierman. Self-dual systems of these lattices are constructed, and the duality of the crossing probability leads to the new exact thresholds. References: 1. R. M. Ziff and C. R. Scullard, J. Phys. A 39, 15083 (2006) 2. J. C. Wierman, J. Phys. A 17, 1525 (1984) 3. Ilya A. Gruzberg, Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory, arXiv:math-ph/0607046}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Ziff, R.M. and Scullard, C.M. and Kleban, P. and Simmons, J.J.H.}, biburl = {https://www.bibsonomy.org/bibtex/27860c06f9225825a2cb9e69e37541ec1/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {9f9737d9189bddf6397e8876e357c6aa}, intrahash = {7860c06f9225825a2cb9e69e37541ec1}, keywords = {ising model percolation potts statphys23 thresholds topic-2 universality}, month = {9-13 July}, timestamp = {2007-06-20T10:16:37.000+0200}, title = {Crossing problems and thresholds in percolation and the Potts model}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1048}, year = 2007 }