Self-avoiding Lévy flights at upper marginal dimensions.

Abstract

We study the node-avoiding (NALF) and path-avoiding extensions of the L\'evy flights in terms of the critical exponents \ensuremath{\nu} and \ensuremath{\gamma} and the leading corrections to scaling, using Monte Carlo simulations with enrichment as the main technique. We focus on the upper marginal dimensions of NALF as predicted by the magnetic analogy where the renormalization-group results should be quantitatively exact for NALF if the method is valid at all. Similarly, we also focus on the boundary between the long-range and short-range behavior of NALF above four dimensions where the renormalization results should again be exact. Thus we investigate the self-avoiding L\'evy flights on hypercubic lattices from d=2 to 6 dimensions and obtain behavior consistent with logarithmic corrections to scaling in the moments of their end-to-end distance distributions. In addition, the effective L\'evy index ${\ensuremath{\mu}}_{\mathrm{eff}}$ is determined from the logarithmic averages of individual step sizes and compared for the two self-avoiding extensions.