Polygons and the Lace Expansion

We give an introduction to the lace expansion for self-avoiding walks, with emphasis on self-avoiding polygons, and with a focus on combinatorial rather than analytical aspects. We derive the lace expansion for self-avoiding walks, and show that this is equivalent to taking the reciprocal of the self-avoiding walk generating function. We list some of the rigorous results for self-avoiding walks and polygons in dimensions d > 4, which have been obtained using the lace expansion. Next we indicate how the lace expansion can be used to enumerate self-avoiding walks in all dimensions, as well as to compute coefficients in the 1/d expansion for the connective constant μ and certain critical amplitudes. We then present some heuristic ideas and numerical results concerning the series analysis of the lace expansion and its relevance for the antiferromagnetic singularity of the susceptibility. Finally, we discuss some of the results for high-dimensional lattice trees that have been obtained using the lace expansion.