Quasi-multiplicativity and Regularity for Metric Graph Gaussian Free Fields

We prove quasi-multiplicativity for critical level-sets of Gaussian free fields (GFF) on the metric graphs Zᵈ Z ~ d ( d≥ 3 d ≥ 3 ). Specifically, we study the probability of connecting two general sets located on opposite sides of an annulus with inner and outer radii both of order N , where additional constraints are imposed on the distance of each set to the annulus. We show that for all d ≥ 3 d ≥ 3 except the critical dimension $$d=6$$ d = 6 , this probability is of the same order as N(6-d) 0 N ( 6 - d ) ∧ 0 (serving as a correction factor) times the product of the two probabilities of connecting each set to the closer boundary of this annulus. The analogue for $$d=6$$ d = 6 is also derived, although the upper and lower bounds differ by a divergent factor of Nᵒ⁽¹⁾ N o ( 1 ) . Notably, it was conjectured by Basu and Sapozhnikov [3] that quasi-multiplicativity without correction factor holds for Bernoulli percolation on Zᵈ Z d when 3≤ d<6 3 ≤ d < 6 and fails when $$d>6$$ d > 6 . In high dimensions (i.e., $$d>6$$ d > 6 ), taking into account the similarity between the metric graph GFF and Bernoulli percolation (which was proposed by Werner [19] and later partly confirmed by the authors (2024)), our result provides support to the conjecture that Bernoulli percolation exhibits quasi-multiplicativity with correction factor N⁶⁻ᵈ N 6 - d . A crucial application of quasi-multiplicativity is proving the existence of the incipient infinite cluster (IIC), which has been completed by Basu and Sapozhnikov [3] for Bernoulli percolation for 3≤ d<6 3 ≤ d <