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

Abstract

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 $1-\exp (-\beta \Vert x-y{\Vert }^{-d-\alpha })$, where $\alpha >0$ is fixed and $\beta \ge 0$ is a parameter. This model is known to have a phase transition at some ${\beta }_c<\infty$ if and only if $\alpha <d$. We study the model in the regime $\alpha \ge d$, in which ${\beta }_c=\infty$, and prove that the susceptibility $\chi (\beta )$ (i.e., the expected volume of the cluster at the origin) satisfies \[ χ(β) = β^{\frac{d}{α- d } - o(1)} \qquad \text{as $\beta \to \infty$ if $\alpha >d$} \qquad \text{and} \qquad e^{e^{ Θ(β) }} \qquad \text{as $\beta \to \infty$ if $\alpha =d$.} \] 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.

17 pages