A discontinuous percolation phase transition on the hierarchical lattice

TL;DR

This work analyzes long-range percolation on the hierarchical lattice and establishes a sharp, dimensionally-appropriate threshold for phase transitions driven by the edge kernel decay. It proves that kernels decaying faster than $|e|^{-2d}\log\log|e|$ yield no transition, kernels decaying exactly at this rate produce a discontinuous transition with a positive infinite-cluster density at criticality, and kernels decaying slower lead to a continuous transition; it further provides a hierarchical analogue of the Imbrie–Newman conjecture giving an exact density at the discontinuous point. A key contribution is a robust renormalization framework and finite-block connection bounds that enable both existence and nonexistence results across scales, including a mixed site-bond percolation analysis. The results illuminate the delicate role of the $\log\log$ factor in enabling discontinuity and offer precise, scheme-dependent formulas for critical densities in the hierarchical setting, with potential implications for Euclidean models. Overall, the paper advances understanding of how long-range interactions yield qualitative changes in percolation on hierarchical structures and provides rigorous tools for multi-scale analysis in this context.

Abstract

For long-range percolation on $\mathbb{Z}$ with translation-invariant edge kernel $J$, it is a classical theorem of Aizenman and Newman (1986) that the phase transition is discontinuous when $J(x,y)$ is of order $|x-y|^{-2}$ and that there is no phase transition at all when $J(x,y)=o(|x-y|^{-2})$. We prove a strengthened version of this theorem for the hierarchical lattice, where the relevant threshold is at $|x-y|^{-2d} \log\log |x-y|$ rather than $|x-y|^{-2}$: There is a continuous phase transition for kernels of larger order, a discontinuous phase transition for kernels of exactly this order, and no phase transition at all for kernels of smaller order. As such, $|x-y|^{-2d} \log\log |x-y|$ is essentially the \emph{only} kernel that produces a discontinuous phase transition. We also prove a hierarchical analogue of the ``$M^2β=1$'' conjecture of Imbrie and Newman (1988), which gives an exact formula for the density of the infinite cluster at the point of discontinuous phase transition and remains open in the Euclidean setting.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 2.1
• Definition 2.2
• Lemma 2.3
• proof : Proof of Theorem \ref{['theo:existence large a']} given Lemma \ref{['lem:renorm ineq']}
• proof : Proof of Lemma \ref{['lem:renorm ineq']}