Logarithmic correction in the critical (d=6) one-arm probability for metric graph GFF

Determine the precise asymptotic behavior of the one-arm probability θ6(N) for the Gaussian free field on the metric graph of Z6, specifically establish whether θ6(N) ∼ N^{-2} [ln(N)]^δ for some δ > 0, and identify the value of δ if this asymptotic holds.

Background

The paper studies connectivity properties (one-arm and crossing probabilities) for sign clusters of the Gaussian free field (GFF) on the metric graph of Zd. For d ≠ 6, sharp asymptotics for the one-arm probability θd(N) are known: θd(N) ∼ N^{-d/2+1} for 3 ≤ d < 6 and θd(N) ∼ N^{-2} for d > 6.

In the critical dimension d = 6, current results provide matching lower and near-matching upper bounds, namely N^{-2} ≲ θ6(N) ≲ N^{-2+varsigma(N)} with varsigma(N) = (ln ln N)/(ln^{1/2} N), leaving the exact asymptotic unresolved. The authors note a conjecture proposing a logarithmic correction to the N^{-2} decay, and they avoid d = 6 in their main results due to added technical complexity.