Ising model on a self-avoiding walk

Fluctuation arguments combined with basic features of the geometry of self-avoiding walks (SAWs) are shown to lead to zero transition temperature T c for the Ising model on a SAW. The Ising model on a non-random fractal representing the essential geometrical features of SAWs is treated exactly by a decimation method. It also leads to T c =0, and to Ising SAW critical behavior different from that on an Ising chain.