From a large percolation component to a giant component on expanders

Definitions and mathematical statement

Let $(G_n)$ be finite connected graphs with $|V(G_n)|\to\infty$, maximum degrees bounded by one constant $D$, and edge expansion uniformly bounded below: some $h>0$ satisfies \[ |\{xy\in E(G_n):x\in A,y\notin A\}|\ge h|A| \quad(0<|A|\le |V(G_n)|/2). \] Perform independent bond percolation with parameter $1/2$ on each $G_n$. Write $K_n(v)$ for the connected open subgraph containing $v$, and $L_n$ for the maximum number of vertices of any open component. Diameters of connected graphs are their intrinsic shortest-path diameters; thus paths used to measure $\operatorname{diam}(K_n(v))$ must use open edges.

Suppose there is a deterministic vertex $v_n\in V(G_n)$ for every $n$ such that \[ \mathbb P_{1/2}\!\left(\operatorname{diam}(K_n(v_n))> \tfrac12\operatorname{diam}(G_n)\right)>\tfrac12. \] Must there exist $\delta>0$ such that $\mathbb P_{1/2}(L_n\ge\delta|V(G_n)|)\to1$?

This spells out the source's "giant component with high probability" as a positive fraction of all vertices, with one fixed fraction along the sequence. The graph family need not be vertex-transitive, and the distinguished vertex need not be typical or randomly chosen. The hypothesis gives only a probability exceeding one half of a large-diameter cluster at one vertex. The conclusion asks for a high-probability macroscopic component somewhere in the graph, at the same parameter $1/2$, without increasing the edge-retention probability.

Short English statement

Does a greater-than-half chance of a large-diameter cluster at one vertex of an expander force a giant cluster with high probability?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 10000044 (AMR-099-0044), 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), Open problem 11.7, PDF page 86. Original source indexed by AMR-099-0044. https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=86.
• 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 I. Benjamini, Euclidean vs. Graph Metric, primary numbered question and conventions. https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf.