The lace expansion was introduced by Brydges and Spencer in 1985 [7] to analyse weakly self-avoiding walks in dimensions d > 4. Subsequently it has been generalised and greatly extended, so that it now applies to a variety of problems of interest in probability theory, statistical physics, and combinatorics, including the strictly self-avoiding walk, lattice trees, lattice animals, percolation, oriented percolation, the contact process, random graphs, and the Ising model. A recent survey is [42].
In this chapter, we give an introduction to the lace expansion for self-avoiding walks, with emphasis on self-avoiding polygons. We focus on combinatorial rather than analytical aspects.
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Clisby, N., Slade, G. (2009). Polygons and the Lace Expansion. In: Guttman, A.J. (eds) Polygons, Polyominoes and Polycubes. Lecture Notes in Physics, vol 775. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9927-4_6
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