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

Abstract

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