Double-exponential susceptibility growth in Dyson's hierarchical model with $|x-y|^{-2}$ interaction

We study long-range percolation on the $d$-dimensional hierarchical lattice, in which each possible edge $\{x,y\}$ is included independently at random with inclusion probability $1-\exp ( -\beta \|x-y\|^{-d-\alpha} )$, where $\alpha>0$ is fixed and $\beta\geq 0$ is a parameter. This model is known to have a phase transition at some $\beta_c<\infty$ if and only if $\alpha<d$. We study the model in the regime $\alpha \geq d$, in which $\beta_c=\infty$, and prove that the susceptibility $\chi(\beta)$ (i.e., the expected volume of the cluster at the origin) satisfies $$\chi(\beta) = \beta^{\frac{d}{\alpha - d } - o(1)} \qquad \text{as $\beta \to \infty$ if $\alpha > d$} \qquad \text{and} \qquad e^{e^{ \Theta(\beta) }} \qquad \text{as $\beta \to \infty$ if $\alpha = d$.}$$ This resolves a problem raised by Georgakopoulos and Haslegrave (2020), who showed that $\chi(\beta)$ grows between exponentially and double-exponentially when $\alpha=d$. Our results imply that analogous results hold for a number of related models including Dyson's hierarchical Ising model, for which the double-exponential susceptibility growth we establish appears to be a new phenomenon even at the heuristic level.