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Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions

Alexander Drewitz et al.

Annals of Probability2026https://doi.org/10.1214/25-aop1775article
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Abstract

We investigate the bond percolation model on transient weighted graphs G induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in G have polynomial volume growth with growth exponent α and that the Green’s function for the random walk on G exhibits a power law decay with exponent ν, in the regime 1≤ν≤α2. In particular, this includes the cases of G=Z3 for which ν=1, and G=Z4 for which ν=α2=2. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance R, like R−ν2+o(1). Our results are, in fact, more precise and yield logarithmic corrections when ν>1 as well as corrections of order loglogR when ν=1. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when ν>1 and essentially optimal when ν=1. This extends previous results from (Invent. Math. 232 (2023) 229–299; Ann. Probab. 48 (2020) 1411–1435).

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https://doi.org/https://doi.org/10.1214/25-aop1775

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@article{alexander2026,
  title        = {{Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions}},
  author       = {Alexander Drewitz et al.},
  journal      = {Annals of Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/25-aop1775},
}

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Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions

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