The logarithmic-corrections conjecture for critical long-range percolation

Let $d\geq 1$, and let $J:\mathbb{Z}^d\times\mathbb{Z}^d\to[0,\infty)$ be a symmetric, integrable, translation-invariant kernel. Suppose that there are constants $0<\alpha<d$ and $c>0$ such that

$$c\|x-y\|^{-d-\alpha}\leq J(x,y)\leq C\|x-y\|^{-d-\alpha}$$

for all distinct $x,y\in\mathbb{Z}^d$. Write $K$ for the cluster of the origin and $\beta_c$ for the critical value. Logarithmic-corrections conjecture. If $d<6$ and $\alpha=d/3$, then the logarithmic corrections to scaling are the same as for the hierarchical lattice. In particular,

$$\mathbb{P}_{\beta_c}(|K|\geq n)\asymp\frac{(\log n)^{1/4}}{\sqrt n}$$

as $n\to\infty$.

This predicts the critical logarithmic corrections at the upper-critical boundary for long-range percolation. Some corresponding corrections have been computed for the hierarchical model, and they differ from those predicted for nearest-neighbor percolation in six dimensions.