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Infinite paths in randomly oriented lattices
Author(s) -
Grimmett Geoffrey R.
Publication year - 2001
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.1007
Subject(s) - mathematics , lattice (music) , square lattice , enhanced data rates for gsm evolution , combinatorics , independence (probability theory) , statistical physics , discrete mathematics , physics , computer science , statistics , telecommunications , acoustics , ising model
Abstract The square lattice is used to generate an oriented graph in which a rightward or upward arrow is present on each edge with probability a , and a leftward or downward arrow with probability b . Independence between different edges of the square lattice is assumed, but nothing is assumed concerning the dependence between the two possible orientations at any given edge. A property of self‐duality is exploited to show that, when a + b =1, the process is, in a sense to be made precise, either critical or supercritical, but not subcritical. This observation enables progress with the percolation problem in which each horizontal edge is oriented rightward with probability p and otherwise leftward, and each vertical edge is oriented upward with probability p and otherwise downward. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 257–266, 2001