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The Self-Avoiding-Walk and Percolation Critical Points in High Dimensions

Published online by Cambridge University Press:  12 September 2008

Takashi Hara  and
Gordon Slade
Takashi Hara
Affiliation:
Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 OEH, UK
Gordon Slade
Affiliation:
Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 OEH, UK
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Abstract

We prove the existence of an asymptotic expansion in the inverse dimension, to all orders, for the connective constant for self-avoiding walks on ℤd. For the critical point, defined as the reciprocal of the connective constant, the coefficients of the expansion are computed through order d−6, with a rigorous error bound of order d−7 Our method for computing terms in the expansion also applies to percolation, and for nearest-neighbour independent Bernoulli bond percolation on ℤd gives the 1/d-expansion for the critical point through order d−3, with a rigorous error bound of order d−4 The method uses the lace expansion.

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Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 4 , Issue 3 , September 1995 , pp. 197 - 215
Copyright
Copyright © Cambridge University Press 1995

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