Isoperimetric dimension and nontrivial percolation threshold

Definitions and mathematical statement

Let $G=(V,E)$ be an infinite connected graph with uniformly bounded degree. For a finite vertex set $A$, let $\partial_G A$ be its external vertex boundary. Its isoperimetric dimension is \[ \operatorname{Dim}(G)=\sup\left\{s\ge1:\exists c>0\ \forall A\subset V, \ 0<|A|<\infty,\quad |\partial_G A|\ge c|A|^{(s-1)/s}\right\}. \] For Bernoulli bond percolation with parameter $p$, independently retain each edge with probability $p$. If $o$ is any fixed vertex, define \[ p_c(G)=\inf\{p\in[0,1]:\mathbb P_p(o\text{ lies in an infinite open component})>0\}. \] Connectedness makes this threshold independent of the root. Prove the implication $\operatorname{Dim}(G)>1\Longrightarrow p_c(G)<1$.

The isoperimetric inequality is required for every finite set, with a constant independent of its position and cardinality. No transitivity, polynomial growth, planarity, or symmetry is assumed. The desired parameter below one may depend on the entire graph; no universal upper bound for all such graphs is requested.

The source separately proposes the weaker case $\operatorname{Dim}(G)=\infty$. That case is included in the established theorem of Duminil-Copin, Goswami, Raoufi, Severo and Yadin: dimension greater than four suffices. Thus the unresolved target retained here is the full implication above one, in particular the range $1<\operatorname{Dim}(G)\le4$. This distinction separates the full target from the already established weaker high-dimensional case.

Short English statement

Does any uniform isoperimetric dimension strictly above one guarantee percolation after deleting a sufficiently small positive fraction of edges?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 10000019 (AMR-099-0019), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Benjamini - Coarse Geometry and Randomness (Saint-Flour notes), Conjecture 4.14, PDF page 34. Original source indexed by AMR-099-0019. https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=34.
• S3 I. Benjamini, Coarse Geometry and Randomness, primary Saint-Flour notes; the numbered question and its surrounding definitions. https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf.
• S4 H. Duminil-Copin et al., Existence of phase transition for percolation using the Gaussian Free Field, Theorem 1.2 (dimension greater than four). https://arxiv.org/abs/1806.07733.