title: Percolation in finite matching lattices.
abstract: We derive an exact, simple relation between the average number of clusters and the wrapping probabilities for two-dimensional percolation. The relation holds for periodic lattices of any size. It generalizes a classical result of Sykes and Essam, and it can be used to find exact or very accurate approximations of the critical density. The criterion that follows is related to the criterion used by Scullard and Jacobsen to find precise approximate thresholds, and our work provides a different perspective on their approach.
authors.affiliation: Institut für Theoretische Physik, Otto-von-Guericke Universität, PF 4120, 39016 Magdeburg, Germany and Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, New Mexico 87501, USA.
authors.forename: Stephan
authors.initials: S
authors.lastname: Mertens
authors.affiliation: Center for the Study of Complex Systems and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA.
authors.forename: Robert M
authors.initials: RM
authors.lastname: Ziff