THE SPIRALING SELF-AVOIDING WALK IN A RANDOM ENVIRONMENT

We study the spiralling self-avoiding lattice walk in a random environment. Upon scaling the temperature appropriately with system size, a phase transition appears. In the low temperature phase the walk segments occupy a few low-energy positions, in the high-temperature phase they are effectively free. An analogy with the random energy model is pointed out. The average size of an N-step walk is shown to be asymptotically proportional to N1/2 log N (as was known for the homogeneous lattice), with a coefficient that increases as the temperature is lowered. The spatial distribution of the walk segments is qualitatively different above and below the critical temperature. The model also allows for a spin glass interpretation, and as such helps to clarify the connection between the concepts of frustration and the chaoticity of the pair correlation both above and below the critical point.