93-286 Hara T., Slade, G.

The Self-Avoiding-Walk and Percolation Critical Points in High Dimensions (531K, PostScript) Nov 5, 93

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Abstract. We prove 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 to be 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. (To potential readers: This is my first trial on Texas archive. Any problems, please notify to hara@ap.titech.ac.jp.)

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