Robert Ziff - University of Michigan
Wednesday, March 4, 2026
Zoom opens: 10:30AM EST
Seminar begins: 10:45AM EST
Exact results and universality in anisotropic percolation
Fundamental concepts in percolation include the phase transition, critical exponents, scaling, and universality. In this talk I will address some of these points especially as related to anisotropic systems.
Exact percolation thresholds are known not just for the well-known cases (square-bond, triangular bond or site, honeycomb bond, Sykes and Essam 1964) but also for broad classes of two-dimensional lattices using self-dual hypergraphs using the “all equals none” criterion of criticality (Ziff and Scullard, 2006, Chayes and Lei, 2006). The isoradial construction approach (Grimmett and Manolescu 2014) can also be used to find thresholds for some additional systems and confirms Wu’s 1979 prediction for the critical manifold of the general checkerboard lattice. Jacobsen and Scullard (2012) extended the all-equals-none idea to non-self-dual systems to develop the critical polynomial approach and a rapidly converging criterion for percolation which yields highly precise estimates of thresholds for many lattices. Mertens and Ziff (2016) extended the Sykes-Essam results for cluster statistics lattices and dual or matching lattices to derive an alternate formulation of the critical-polynomial approach based on cluster numbers, leading to a simple algorithm generalizing the Newman-Ziff (2000) approach that can be used to find thresholds easily and efficiently. The isoradial result relates anisotropic percolation to an isotropic system of a different dimension (Grimmett and Manolescu, 2014, Kovacs, Igloi and Cardy, 2013), and we verify that excess cluster numbers (Kleban and Ziff, 1998) and various Binder coefficients (Binder, 1981, Hu Blote and Deng 2012) have universal properties when the anisotropic system shape is properly scaled.. We also examine the cluster numbers with the result of Temperley and Lieb, 1971 generalized for a general anisotropic system (Scullard and Ziff, to be published). There may be implications to the enclosed area distribution (Cardy and Ziff, 2002) for anisotropic percolation as well — an area for future study.