%0 Book Section %1 statphys23_0897 %A Kleban, P. %A Simmons, J.J.H. %A Ziff, R.M. %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 conformal correlations field percolation point statphys23 theory topic-1 %T Exact Factorization of Correlations in 2-D Critical Percolation %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=897 %X Using conformal field theory, we calculate several connection probabilities for 2-D systems in the half-plane (or any conformally equivalent geometry) at the percolation point 1. We find explicit expressions for these quantities (up to non-universal multiplicative constants). In addition, they may be expressed in terms of each other. For example, equation P(z,x_1,x_2) = C\;P(x_1,x_2)P(z,x_1)P(z,x_2), equation where $P$ is the probability density of finding a cluster that connects its arguments. Here $x_i$ are points on the real axis (the boundary), while $z$ is in the interior. The constant, given by 2 equation C=2^7/2\pi^5/23^3/4\Gamma(1/3)^9/2 = 1.0299268łdots equation is a boundary operator product expansion coefficient. Note that (1) and (2) are both exact and universal, i.e.\ independent of the particular realization of percolation. Due to conformal invariance, they hold in any simply-connected region--the individual probabilities in (1) change, but the relation between them remains the same. We verified (1) by high-precision computer simulation on square systems 1, as illustrated. In the figure, the upper panels show simulated values for $P(x_i,z)$ for $i=1,2$. The lower left panel shows simulated values for $P(x_1,x_2,z)/P(x_1,x_2)$ while the lower right panel shows simulated values for $P(z,x_1)P(z,x_2)/P(x_1,x_2)$, finally multiplied by a constant $C$ to make the contours agree quantitatively with those of the lower left panel. This gives $C=1.030 0.001$, in excellent agreement with (2). This work was supported in part by the National Science Foundation Grants Nos. DMR-0536927 (PK) and DMS-0553487 (RMZ) .\\ 1) Peter Kleban, Jacob J. H. Simmons, and Robert M. Ziff, Anchored Critical Percolation Clusters and 2-D Electrostatics, Phys. Rev. Letters 97 (2006) 115702 arXiv: cond-mat/0605120.\\ 2) Jacob J. H. Simmons and Peter Kleban, preprint.

@incollection{statphys23_0897, abstract = {Using conformal field theory, we calculate several connection probabilities for 2-D systems in the half-plane (or any conformally equivalent geometry) at the percolation point [1]. We find explicit expressions for these quantities (up to non-universal multiplicative constants). In addition, they may be expressed in terms of each other. For example, \begin{equation} {\cal P}(z,x_1,x_2) = C\;\sqrt{{\cal P}(x_1,x_2){\cal P}(z,x_1){\cal P}(z,x_2)}, \end{equation} where ${\cal P}$ is the probability density of finding a cluster that connects its arguments. Here $x_i$ are points on the real axis (the boundary), while $z$ is in the interior. The constant, given by [2] \begin{equation} C=\frac{2^{7/2}\pi^{5/2}}{3^{3/4}\Gamma(1/3)^{9/2}} = 1.0299268\ldots \end{equation} is a boundary operator product expansion coefficient. Note that (1) and (2) are both exact and universal, i.e.\ independent of the particular realization of percolation. Due to conformal invariance, they hold in any simply-connected region--the individual probabilities in (1) change, but the relation between them remains the same. We verified (1) by high-precision computer simulation on square systems [1], as illustrated. In the figure, the upper panels show simulated values for ${\cal P}(x_i,z)$ for $i=1,2$. The lower left panel shows simulated values for ${\cal P}(x_1,x_2,z)/{\cal P}(x_1,x_2)$ while the lower right panel shows simulated values for $\sqrt{{\cal P}(z,x_1){\cal P}(z,x_2)/{\cal P}(x_1,x_2)}$, finally multiplied by a constant $C$ to make the contours agree quantitatively with those of the lower left panel. This gives $C=1.030 \pm 0.001$, in excellent agreement with (2). This work was supported in part by the National Science Foundation Grants Nos. DMR-0536927 (PK) and DMS-0553487 (RMZ) .\\ 1) Peter Kleban, Jacob J. H. Simmons, and Robert M. Ziff, {\it Anchored Critical Percolation Clusters and 2-D Electrostatics}, Phys. Rev. Letters {\bf 97} (2006) 115702 [arXiv: cond-mat/0605120].\\ 2) Jacob J. H. Simmons and Peter Kleban, {\it preprint}.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Kleban, P. and Simmons, J.J.H. and Ziff, R.M.}, biburl = {https://www.bibsonomy.org/bibtex/21e89005e46121514d146a3d4ea927eae/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {3339fafa8a000b8da076c11c8012a62d}, intrahash = {1e89005e46121514d146a3d4ea927eae}, keywords = {conformal correlations field percolation point statphys23 theory topic-1}, month = {9-13 July}, timestamp = {2007-06-20T10:16:33.000+0200}, title = {Exact Factorization of Correlations in 2-D Critical Percolation}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=897}, year = 2007 }