Quasi-multiplicativity and Regularity for Metric Graph Gaussian Free Fields
- 1New Cornerstone Science Laboratory, School of Mathematical Sciences, Peking University, No. 5 Yiheyuan Road, Beijing, 100871, China
- 2Faculty of Mathematics and Computer Science, Weizmann Institute of Science, No. 234 Herzl Street, 7610001, Rehovot, Israel
Abstract
We prove quasi-multiplicativity for critical level-sets of Gaussian free fields (GFF) on the metric graphs ℤ̃d 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 <
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BibTeX
@article{ding2026quasimultiplicativity,
title = {{Quasi-multiplicativity and Regularity for Metric Graph Gaussian Free Fields}},
author = {Jian Ding and Zhenhao Cai},
journal = {Communications in Mathematical Physics},
year = {2026},
month = oct,
doi = {10.1007/s00220-026-05772-5},
url = {https://doi.org/10.1007/s00220-026-05772-5},
note = {CC BY},
}
RIS
TY - JOUR
TI - Quasi-multiplicativity and Regularity for Metric Graph Gaussian Free Fields
AU - Jian Ding
AU - Zhenhao Cai
PY - 2026
DA - 2026/10/28
JO - Communications in Mathematical Physics
DO - 10.1007/s00220-026-05772-5
UR - https://doi.org/10.1007/s00220-026-05772-5
AB - We prove quasi-multiplicativity for critical level-sets of Gaussian free fields (GFF) on the metric graphs ℤ̃^d 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 N^o(1) N o ( 1 ) . Notably, it was conjectured by Basu and Sapozhnikov [3] that quasi-multiplicativity without correction factor holds for Bernoulli percolation on ℤ^d 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 N^6−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 <
SN - 0010-3616
SN - 1432-0916
ER -