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

Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation - Mathematics > Probability - Download this document for free, or read online. Document in PDF available to download.

Abstract: We consider random walk and self-avoiding walk whose 1-step distribution isgiven by $D$, and oriented percolation whose bond-occupation probability isproportional to $D$. Suppose that $Dx$ decays as $|x|^{-d-\alpha}$ with$\alpha>0$. For random walk in any dimension $d$ and for self-avoiding walk andcritical-subcritical oriented percolation above the common upper-criticaldimension $d {\mathrm{c}}\equiv2\alpha\wedge2$, we prove large-$t$asymptotics of the gyration radius, which is the average end-to-end distance ofrandom walk-self-avoiding walk of length $t$ or the average spatial size of anoriented percolation cluster at time $t$. This proves the conjecture forlong-range self-avoiding walk in Ann. Inst. H. Poincar\-{e} Probab. Statist.2010, to appear and for long-range oriented percolation in Probab. TheoryRelated Fields 142 2008 151-188 and Probab. Theory Related Fields 1452009 435-458.

Author: ** Lung-Chi Chen, Akira Sakai**

Source: https://arxiv.org/