Double-exponential susceptibility growth in Dyson's hierarchical model with $\mathrm{\mid }x-y{\mathrm{\mid }}^{-2}$ 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 $1-\exp (-\beta \mathrm{\parallel }x-y{\mathrm{\parallel }}^{-d-\alpha })$, where $\alpha >0$ is fixed and $\beta \ge 0$ is a parameter. This model is known to have a phase transition at some ${\beta }_c<\mathrm{\infty }$ if and only if $\alpha <d$. We study the model in the regime $\alpha \ge d$, in which ${\beta }_c=\mathrm{\infty }$, and prove that the susceptibility $\chi (\beta )$ (i.e., the expected volume of the cluster at the origin) satisfies $$\chi (\beta )={\beta }^{\frac{d}{\alpha -d}-o(1)}\text{as }\beta \to \mathrm{\infty }\text{ if }\alpha >d\text{and}{e}^{e^{\mathrm{\Theta }(\beta )}}\text{as }\beta \to \mathrm{\infty }\text{ if }\alpha =d\text{.}$$ 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.