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# High-Dimensional Graphical Networks of Self-Avoiding Walks

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Published:2004-02-01
Printed: Feb 2004
• Mark Holmes
• Antal A. Járai
• Akira Sakai
• Gordon Slade
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## Abstract

We use the lace expansion to analyse networks of mutually-avoiding self-avoiding walks, having the topology of a graph. The networks are defined in terms of spread-out self-avoiding walks that are permitted to take large steps. We study the asymptotic behaviour of networks in the limit of widely separated network branch points, and prove Gaussian behaviour for sufficiently spread-out networks on $\mathbb{Z}^d$ in dimensions $d>4$.
 MSC Classifications: 82B41 - Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 60K35 - Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

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