TAILIEUCHUNG - Báo cáo toán học: "Aperiodic non-isomorphic lattices with equivalent percolation and random-cluster models"

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: Aperiodic non-isomorphic lattices with equivalent percolation and random-cluster models. | Aperiodic non-isomorphic lattices with equivalent percolation and random-cluster models. Klas Markstrom and John C. Wierman Submitted May 6 2009 Accepted Mar 2010 Published Mar 29 2010 Mathematics Subject Classification 60K35 05C80 05C65 Abstract We explicitly construct an uncountable class of infinite aperiodic plane graphs which have equal and explicitly computable bond percolation thresholds. Furthermore for both bond percolation and the random-cluster model all large scale properties such as the values of the percolation threshold and the critical exponents of the graphs are equal. This equivalence holds for all values of p and all q E 0 to for the random-cluster model. The graphs are constructed by placing a copy of a rotor gadget graph or its reflection in each hyperedge of a connected self-dual 3-uniform plane hypergraph lattice. The exact bond percolation threshold may be explicitly determined as the root of a polynomial by using a generalised star-triangle transformation. Related randomly oriented models share the same bond percolation threshold value. 1 Introduction and Summary Although it has been over fifty years since the origins of percolation theory until recently the percolation threshold had been exactly determined for only a small collection of lattice graphs. Periodicity has been a crucial assumption for rigorous proofs of all these exact percolation threshold values. Research of Scullard 10 Ziff 20 Scullard and Ziff 11 and Ziff and Scullard 21 used a generalised star-triangle transformation and a concept of triangle-duality to predict exact solutions for a collection of periodic lattice graphs in two dimensions. Wierman and Ziff 19 organised this approach to describe a construction of an infinite class of planar graphs for which the bond percolation threshold can be mathematically rigorously determined. The graphs are constructed by placing isomorphic copies of a finite connected planar graph with three terminals called a generator in a .

• Báo cáo khoa học
• báo cáo của tạp chí Department of Mathematic
• tài liệu báo cáo nghiên cứu khoa học
• cách trình bày báo cáo
• kiến thức toán học
• báo cáo toán học