Self-avoiding Lévy flights in one dimension.

Abstract

We study the node-avoiding (NALF) and path-avoiding (PALF) extensions of the L\'evy flights in one dimension both numerically and analytically. The asymptotic behavior of the PALF is determined exactly while that of NALF has been available from the mapping to a spin model. Monte Carlo results for both types of self-avoiding L\'evy flights are used to study the convergence toward the asymptotic behavior. We find very large corrections to asymptotic scaling in NALF for a wide range of the L\'evy index \ensuremath{\mu} and also, surprisingly, that the moments of the end-to-end distance of the NALF are greater than those of the PALF when they both exist. Based on these observations we conclude that the morphology of the NALF is far more complex than that of the PALF or the random L\'evy flights, and that the NALF and PALF are certainly in different universality classes in one dimension.