2 problems

Let $d\geq 3$ and consider the long-range percolation graph on $\mathbb Z^d$ with connection probabilities having the asymptotic decay specified by the paper for some $s>d+2$. A qu…

Let $d=4$, and let $\mathbb P$ be any i.i.d. environment law satisfying $\mathbb P(0<\omega_b\leq 1)=1$. Write $\cmss P_\omega^{2n}(0,0)$ for the quenched return probability at tim…