Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Self-dual Planar Hypergraphs and Exact Bond Percolation Thresholds. | Self-dual Planar Hypergraphs and Exact Bond Percolation Thresholds John C. Wierman Department of Applied Mathematics and Statistics Johns Hopkins University wierman@ Robert M. Ziff Michigan Center for Theoretical Physics and Department of Chemical Engineering University of Michigan rziff@ Submitted Oct 4 2010 Accepted Mar 6 2011 Published Mar 24 2011 Mathematics Subject Classification 05C65 60K35 82B43 Abstract A generalized star-triangle transformation and a concept of triangle-duality have been introduced recently in the physics literature to predict exact percolation threshold values of several lattices. Conditions for the solution of bond percolation models are investigated and an infinite class of lattice graphs for which exact bond percolation thresholds may be rigorously determined as the solution of a polynomial equation are identified. This class is naturally described in terms of hypergraphs leading to definitions of planar hypergraphs and self-dual planar hypergraphs. There exist infinitely many self-dual planar 3-uniform hypergraphs and as a consequence there exist infinitely many real numbers a E 0 1 for which there are infinitely many lattices that have bond percolation threshold equal to a. 1 Introduction Bond Percolation Percolation is a random model on infinite lattices which serves as the simplest lattice model example of a process exhibiting a phase transition. Even so it provides some ex Research supported by the Acheson J. Duncan Fund for the Advancement of Research in Statistics at Johns Hopkins University and by sabbatical funding from the Mittag-Leffler Institute of the Swedish Royal Academy of Sciences Research supported by National Science Foundation Grant No. DMS-0553487 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P61 1 tremely challenging problems. Its study provides intuition for more elaborate statistical mechanics models. Due to its focus on clustering and connectivity phenomena it is applied widely to problems
