Exact order of the critical one-arm probability at dimension d=6

Determine the exact asymptotic order of the critical one-arm probability θ_6(N) for the Gaussian free field on the metric graph of Z^6, where θ_d(N) = P(0 ↔ ∂B(N)) at level 0. Current results give N^{-2} ≲ θ_6(N) ≲ N^{-2+varsigma(N)} with varsigma(N) = (ln ln N)/(ln^{1/2} N), and it has been conjectured that θ_6(N) is of the form N^{-2} [ln(N)]^δ for some δ > 0.

Background

The paper studies connectivity properties of sign clusters of the Gaussian free field (GFF) on the metric graph of Zd at the critical level 0. A central quantity is the one-arm probability θ_d(N) = P(0 ↔ ∂B(N)), for which precise asymptotics are known for d ≠ 6: θ_d(N) ≍ N{-d/2+1} for 3 ≤ d < 6 and θ_d(N) ≍ N{-2} for d > 6.

At the critical dimension d = 6, only bounds are available: N{-2} ≲ θ_6(N) ≲ N{-2+varsigma(N)} with varsigma(N) = (ln ln N)/(ln{1/2} N). The exact order at d=6 remains unknown, and prior work conjectures a logarithmic correction of the form N{-2} [ln(N)]δ for some δ > 0. Resolving this would complete the picture of one-arm exponents across dimensions.

References

The exact order of θ6(N) remains open, and it has been conjectured to be N{-2}[\ln(N)]\delta for some \delta>0 (see Remark 1.5).

— Heterochromatic two-arm probabilities for metric graph Gaussian free fields (2510.20492 - Cai et al., 23 Oct 2025) in Section 1 (Introduction), after equation (one_arm_6)