Critical long-range percolation II: Low effective 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. We focus on the long-range low-dimensional (LR-LD) regime $d/3<\alpha<\alpha_c(d)$, where the model is below its upper critical dimension. Since computing $\alpha_c(d)$ for $2<d<6$ appears to be beyond the scope of current techniques, we give an axiomatic definition of the LR regime which we prove holds for $\alpha <1$. Using this, we prove up-to-constants estimates for the critical and slightly subcritical two-point function in the LR regime and for the volume tail and $k$-point function in the LR-LD regime. We deduce that the critical exponents satisfy the identities $$\eta = 2-\alpha, \qquad \gamma = (2-\eta)\nu, \qquad \text{ and } \qquad \Delta = \nu d_f$$ in the LR regime (if $\gamma$, $\nu$, or $\Delta$ is well-defined) and that $\delta$ and $d_f$ follow the hyperscaling identities $$\delta = \frac{d+\alpha}{d-\alpha} \qquad \text{ and } \qquad d_f = \frac{d+\alpha}{2}$$ in the LR-LD regime. Our results are suggestive of conformal invariance in the LR-LD regime, with the critical $k$-point function matching an explicit M\"obius-covariant function up-to-constants.