Long-range self-avoiding walk converges to alpha-stable processes

Abstract

We consider a long-range version of self-avoiding walk in dimension $d > 2(\alpha \wedge 2)$, where $d$ denotes dimension and $\alpha$ the power-law decay exponent of the coupling function. Under appropriate scaling we prove convergence to Brownian motion for $\alpha \ge 2$, and to $\alpha$-stable L\'evy motion for $\alpha < 2$. This complements results by Slade (1988), who proves convergence to Brownian motion for nearest-neighbor self-avoiding walk in high dimension.