Abstract.
The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den Hollander and the authors. We give conditions under which the solution to the recursion relation behaves as a Gaussian, both in Fourier space and in terms of a local central limit theorem. These conditions are shown elsewhere to hold for sufficiently spread-out models of networks of self-avoiding walks in dimensions d > 4, and for sufficiently spread-out models of critical oriented percolation in dimensions d + 1 > 5, providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these models.
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Received: 13 September 2000 / Revised version: 16 May 2001 / Published online: 20 December 2001
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van der Hofstad, R., Slade, G. A generalised inductive approach to the lace expansion. Probab Theory Relat Fields 122, 389–430 (2002). https://doi.org/10.1007/s004400100175
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DOI: https://doi.org/10.1007/s004400100175