Long-range self-avoiding walk converges to α-stable processes
We consider a long-range version of self-avoiding walk in dimension > 2( ∧ 2), where denotes dimension and the power-law decay exponent of the coupling function. Under appropriate scaling we prove convergence to brownian motion for ≥ 2, and to -stable Lévy motion for < 2. This complements results by Slade [ (1988) L417–L420], who proves convergence to brownian motion for nearest-neighbor self-avoiding walk in high dimension.