Duminil-Copin Hugo, Manolescu Ioan, Tassion Vincent
Université de Genève, Geneva, Switzerland.
Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France.
Probab Theory Relat Fields. 2021;181(1-3):401-449. doi: 10.1007/s00440-021-01060-6. Epub 2021 Jun 19.
This paper is studying the critical regime of the planar random-cluster model on with cluster-weight . More precisely, we prove which are uniform in their boundary conditions and depend only on their extremal lengths. They imply in particular that any fractal boundary is touched by macroscopic clusters, uniformly in its roughness or the configuration on the boundary. Additionally, they imply that . We also obtain a number of properties of so-called arm-events: (two arms in the half-plane, three arms in the half-plane and five arms in the bulk), and properties (even when arms are not alternating between primal and dual), and the fact that the . These results were previously known only for Bernoulli percolation ( ) and the FK-Ising model ( ). Finally, we prove new bounds on the one, two and four-arm exponents for , as well as the one-arm exponent in the half-plane. These improve the previously known bounds, even for Bernoulli percolation.
本文研究具有簇权重的平面随机簇模型的临界状态。更确切地说，我们证明了在其边界条件下是一致的，并且仅取决于它们的极值长度。特别地，这意味着任何分形边界都被宏观簇触及，与其粗糙度或边界上的构型无关。此外，这意味着 。我们还获得了一些所谓臂事件的性质：（半平面中的两条臂、半平面中的三条臂和体中的五条臂），以及 性质（即使臂在原对偶之间不交替），以及 的事实。这些结果以前仅在伯努利渗流（ ）和FK - 伊辛模型（ ）中已知。最后，我们证明了 中一臂、二臂和四臂指数的新界，以及半平面中的一臂指数。即使对于伯努利渗流，这些也改进了先前已知的界。
