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Drewitz, Alexander ORCID: 0000-0002-5546-3614, Prévost, Alexis ORCID: 0000-0001-7273-0481 and Rodriguez, Pierre-François (2025). Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions. Arxiv. pp. 1-27. Cornell University.

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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 ď ν ď α 2 . In particular, this includes the cases of G “ Z3 for which ν “ 1, and G “ Z4 for which ν “ α 2 “ 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´ ν 2 `op1q. Our results are, in fact, more precise and yield logarithmic corrections when ν ą 1 as well as corrections of order log log R when ν “ 1. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when ν ą 1 and essentially optimal when ν “ 1. This extends previous results from [16, 12]

Item Type:	Article
Creators:	Creators Email ORCID ORCID Put Code Drewitz, Alexander UNSPECIFIED https://orcid.org/0000-0002-5546-3614 UNSPECIFIED Prévost, Alexis UNSPECIFIED https://orcid.org/0000-0001-7273-0481 UNSPECIFIED Rodriguez, Pierre-François UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-799497
Identification Number:	10.48550/arXiv.2312.10030
Journal or Publication Title:	Arxiv
Page Range:	pp. 1-27
Date:	31 October 2025
Publisher:	Cornell University
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Institute of Computer Science
Subjects:	Data processing Computer science Mathematics
['eprint_fieldname_oa_funders' not defined]:	Publikationsfonds UzK
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/79949