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Critical points and intermediate phases on wedges of $${\mathbb{Z}}$$ d. (English) Zbl 0636.60105
This paper investigates the phase structure of self-avoiding walks, the Ising magnet and bond percolation defined on subsets of $${\mathbb{Z}}^ d,$$ $$d\geq 2$$, which are shaped like ‘wedges’. There was an apparent inconsistency between results of J. M. Hammersley and S. G. Whittington [Self-avoiding walks in wedges, ibid. 18, 101-112 (1985)] on the connectivity constant for self-avoiding walks and earlier results of G. R. Grimmett [ibid. 16, 599-604 (1983; Zbl 0508.60082), and Adv. Appl. Probab. 13, 314-324 (1981; Zbl 0459.60098)] on bond percolation in $${\mathbb{Z}}^ 2.$$
The present paper explains this phenomenon, and extends these results to a more general class of models; and thus shows that there are unusual phase structures for the Ising magnet and bond percolation in certain edges.
Reviewer: C.McDiarmid

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G50 Sums of independent random variables; random walks
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