The lace expansion is one of the few methods to rigorously prove critical behavior for various models in high dimensions. It was initiated by David Brydges and Thomas Spencer in 1985 to show degeneracy of the critical behavior for weakly self-avoiding walk in d > 4 dimensions to that for random walk. This is one of the reasons Brydges was awarded the Poincaré Prize at ICMP this summer. Self-avoiding walk is a standard model for linear polymers in a good solvent. Other models to which the lace expansion has been successfully applied are percolation, lattice trees/animals, the contact process, the Ising and phi^4 models. In the colloquium talk, I will explain the lace expansion for self-avoiding walk (past) and how it has been extended to other models listed above (present). If time permits, I will also explain the current status of my ongoing work on the lace expansion for self-avoiding walk on random conductors (future).
