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Exact value of the giant-component threshold radius r_g(d) in continuum percolation

Determine the exact value of the critical radius r_g(d) for continuum percolation on R^d with unit-intensity Poisson Point Process under the Gilbert disk graph (random geometric graph) model, where r_g(d) is defined as the infimum r ≥ 0 such that the origin belongs to an infinite connected component with positive probability, for each dimension d ≥ 2.

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Background

The paper studies random geometric graphs generated by a unit-intensity Poisson Point Process and defines r_g(d) as the percolation threshold radius for the existence of an infinite component in the infinite-volume model G{r,d}. While it is established that 0 < r_g(d) < ∞ for every d ≥ 2, its exact value remains undetermined. The authors reference known estimates in dimension d = 2 but emphasize the absence of an exact determination in general.

References

"It is well known that $0<r_g<\infty$ for every $d\ge 2$, although its exact value is not known (see for estimates in $d=2$)."

— Mixing time and isoperimetry in random geometric graphs (2510.19951 - Kiwi et al., 22 Oct 2025) in Section 1 (Introduction), definition of r_g after introducing G^{r,d}