| Abstract: |
| Consider the long-range self-avoiding walks on $\mathbb{Z}^d$, whose one step distribution $D(x)$ decays as $|x|^{-d-\alpha}$ for some $\alpha>0$. In our previous work (2015), we have shown that, for $\alpha\neq 2$, the critical two-point function $G_{p_c} (x)$ decays as $|x|^{\alpha\wedge 2-d}$ above the upper-critical dimension $d_c :=2 (\alpha\wedge 2)$. In this talk, we show that $G_{p_c} (x)$ for $\alpha = 2$ decays as$ |x|^{2-d}/ \log |x|$ whenever $d \geq d_c$ (including equality). This solves the conjecture in (2015), extended all the way down to $d = d_c$, and confirms a part of predictions in physics (2014). The proof relies on the lace expansion and new convolution bounds on power functions with log corrections. |
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