Node-Avoiding Lévy Flight: A Numerical Test of theεExpansion

We study an extension of Lévy flight to include self-repulsion in the path of the walk. We call the extension node-avoiding Lévy flight and we show its equivalence to the $n\to 0$ limit of a statistical mechanical model for a magnetic system with long-range interactions between the spins. By use of this equivalence we are able to make a detailed comparison between the results of the $\epsilon$ expansion for the magnetic model, a Monte Carlo simulation of the Lévy flight model, and the results of a Flory-type argument. This is the first comparison of the $\epsilon$ expansion for $\epsilon \ll 1$ with a numerical simulation for any model. Some speculations are made on applications of the model of node-avoiding Lévy flight.