國立清華大學 數學系

Abstract We consider long-range self-avoiding walk on $\mathbb{Z}^d$ whose one-step distribution is given by $D$. Suppose $D(x)$ decays as $|x|^{-d-\alpha}$ with $\alpha>0$. In this talk, I present that for $d>2(\alpha\wedge 2)$ and the spread-out parameter sufficiently large, the critical two-point function $G_{p_c}(x)$ for each model is asymptotically $\frac {C}{|x|^{d-\alpha\wedge 2}}$ if $\alpha\neq 2$ and $\frac {C}{|x|^{d-2}\log |x|}$ if $\alpha=2$, where the constant $C\in(0,\infty)$ is expressed in terms of the lace-expansion coefficients and exhibits crossover between $\alpha< 2$ and $\alpha > 2$. Note that the upper-critical dimension for long-range self-avoiding walk is $2(\alpha\wedge2)$. The talk is based on a work joint with Akira Sakai.