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We study long-range percolation on the $d$-dimensional hierarchical lattice,in which each possible edge $\{x,y\}$ is included independently at random withinclusion 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 phasetransition at some $\beta_c<\infty$ if and only if $\alpha<d$. We study themodel in the regime $\alpha \geq d$, in which $\beta_c=\infty$, and prove thatthe susceptibility $\chi(\beta)$ (i.e., the expected volume of the cluster atthe 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 anddouble-exponentially when $\alpha=d$. Our results imply that analogous resultshold for a number of related models including Dyson's hierarchical Ising model,for which the double-exponential susceptibility growth we establish appears tobe a new phenomenon even at the heuristic level.

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