Critical points and intermediate phases on wedges of Zd

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Abstract
The authors examine the phase structure of self-avoiding walks, the Ising magnet and bond percolation defined on subsets of Zd, d>or=2, which have the geometry of wedges. They prove that if the cross sectional area of the wedge diverges with its width, then the high temperature critical point of any of these models defined on the wedge coincides with that of the corresponding model on the full lattice. For the Ising magnet and bond percolation, they show that there is a non-trivial low temperature critical point if the cross sectional area of the wedge diverges logarithmically with its width. Moreover, for bond percolation they show that the low temperature critical point of a logarithmic wedge may be made arbitrarily close to that of the full lattice by taking the coefficient of the logarithm large enough. Corollaries to these theorems include the existence of an intermediate phase for the Ising magnet and bond percolation on logarithmic wedges and the existence of a first-order transition for percolation models on a subclass of these wedges.

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