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

TL;DR

This paper analyzes the growth of susceptibility in long-range percolation on hierarchical lattices, revealing a double-exponential growth at critical parameters, which extends understanding of phase transitions in related models like Dyson's hierarchical Ising model.

Contribution

The paper proves the precise asymptotic behavior of susceptibility in long-range hierarchical percolation for lpha eld, demonstrating double-exponential growth at the critical point, a novel finding.

Findings

Susceptibility grows polynomially for lpha eld as eta ftarrow ty

Susceptibility exhibits double-exponential growth at lpha = d

Results extend to related models like Dyson's hierarchical Ising model

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 $1-\exp ( -\beta \|x-y\|^{-d-\alpha} )$, 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.