Critical long-range percolation III: The upper critical dimension

In long-range percolation on $\mathbb{Z}^d$, points $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed and $\beta \geq 0$ is a parameter. As $d$ and $\alpha$ vary, the model is conjectured to exhibit eight qualitatively different second-order critical behaviours, with a transition between mean-field and low-dimensional regimes when $d=\min\{6,3\alpha\}$, a transition between long- and short-range regimes at a crossover value $\alpha_c(d)$, and with various logarithmic corrections at the boundaries between these regimes. This is the second of three papers developing a rigorous theory of the model's critical behavior in five of these eight regimes, including all long-range (LR) and high-dimensional (HD) regimes. Here, we analyze the model at its upper critical dimension $d=3\alpha<6$. We prove the hydrodynamic condition holds, which allows us to apply our first paper's RG analysis to deduce that the model has the same superprocess scaling limits as in high dimension, after accounting for slowly varying corrections to scaling. We then compute the precise logarithmic corrections to scaling by analyzing the RG flow to second order. Our results yield in particular that for $d=3\alpha < 6$ the critical volume tail is $$\mathbb{P}_{\beta_c}(|K|\geq n) \sim C \frac{(\log n)^{1/4}}{\sqrt{n}}$$ as $n\to \infty$, while the critical two- and three-point functions are $$\mathbb{P}_{\beta_c}(x\leftrightarrow y) \asymp \|x-y\|^{-d+\alpha} \; \text{ and } \; \mathbb{P}_{\beta_c}(x\leftrightarrow y \leftrightarrow z) \asymp \sqrt{\frac{\|x-y\|^{-d+\alpha}\|y-z\|^{-d+\alpha}\|z-x\|^{-d+\alpha}}{\log(1+\min\{\|x-y\|,\|y-z\|,\|z-x\|\})}}.$$ These logarithmic corrections match those in hierarchical percolation but differ from those conjectured for nearest-neighbour percolation on $\mathbb{Z}^6$.