Abstract:
I will present the method of cluster expansions of "polymer" models, used widely in the statistical mechanics of lattice systems. The method has a long history, and various methods were applied to the estimates of the terms of these series. It seems that the combinatorial aproach to the problem is rather powerful if applied carefully. Substantial simplifications and clarifications of the method were done in recent years. A closer inspection of the cancellations in the cluster expansion series gives formulas, summations over closed paths in the lattice whose usefulness can be tested for classical examples like Kramers - Wannier - Onsager solution and determinants of Laplacians (and their perturbations). The method gives an interesting insight even into elementary formulas like the expansion of log(1+x).