 00468 Takashi Hara, Remco van der Hofstad and Gordon Slade
 Critical twopoint functions and the lace expansion for spreadout
highdimensional percolation and related models
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Nov 28, 00

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Abstract. We consider spreadout models of selfavoiding walk, bond percolation,
lattice trees and bond lattice animals on the ddimensional hyper cubic
lattice having long finiterange connections, above their upper critical
dimensions d=4 (selfavoiding walk), d=6 (percolation) and d=8 (trees and
animals). The twopoint functions for these models are respectively the
generating function for selfavoiding walks from the origin to x, the
probability of a connection from 0 to x, and the generating function for
lattice trees or lattice animals containing 0 and x. We use the lace
expansion to prove that for sufficiently spreadout models above the upper
critical dimension, the twopoint function of each model decays, at the
critical point, as a multiple of $x^{2d}$ as x goes to infinity. We use
a new unified method to prove convergence of the lace expansion. The
method is based on xspace methods rather than the Fourier transform. Our
results also yield unified and simplified proofs of the bubble condition
for selfavoiding walk, the triangle condition for percolation, and the
square condition for lattice trees and lattice animals, for sufficiently
spreadout models above the upper critical dimension.
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