Random Walk Representations of the Heisenberg Model

In this paper we develop some random walk representations for the spin-S Heisenberg ferromagnet with nearest neighbor interaction on the integer lattice in d dimensions. The representations parallel corresponding representations for the perfect Bose gas. Hence the Heisenberg model can be thought of as an interacting Bose gas. This has been made more explicit in the paper of Dyson (5/ (see also ref. 4). In our picture one thinks of the free Bose gas as an ensemble of noninteracting random walks. The corresponding picture for the Heisenberg model is of an ensemble of interacting walks. The effect of the interaction is to slow down the diffusion process. One can be more precise and identify the local time for the Heisenberg diffusion. It is given in terms of the spin-S Ising model. In particular, expectations for the spin-S Heisenberg model conditioned on zero diffusion are just spin-S Ising expectations. Since the effect of diffusion should be to lower critical temperature, one is tempted to conjecture that