Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions
Alexander Drewitz et al.
What the paper says
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).
Evidence weight
Balanced mode · F 0.40 / M 0.15 / V 0.05 / R 0.40
| F · citation impact | 0.50 × 0.4 = 0.20 |
| M · momentum | 0.50 × 0.15 = 0.07 |
| V · venue signal | 0.50 × 0.05 = 0.03 |
| R · text relevance † | 0.50 × 0.4 = 0.20 |
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