Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation

Abstract

We consider random walk and self-avoiding walk whose 1-step distribution is given by $D$, and oriented percolation whose bond-occupation probability is proportional to $D$. Suppose that $D(x)$ decays as $|x{|}^{-d-\alpha }$ with $\alpha >0$. For random walk in any dimension $d$ and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension $d_{\mathrm{c}}\equiv 2(\alpha \wedge 2)$, we prove large-$t$ asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length $t$ or the average spatial size of an oriented percolation cluster at time $t$. This proves the conjecture for long-range self-avoiding walk in [Ann. Inst. H. Poincaré Probab. Statist. (2010), to appear] and for long-range oriented percolation in [Probab. Theory Related Fields 142 (2008) 151--188] and [Probab. Theory Related Fields 145 (2009) 435--458].

Published in at http://dx.doi.org/10.1214/10-AOP557 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk
• Diagrammatic bounds on the lace-expansion coefficients for oriented percolation