Universität zu Köln

Drewitz, Alexander ORCID: 0000-0002-5546-3614, Prévost, Alexis ORCID: 0000-0001-7273-0481 and Rodriguez, Pierre-François ORCID: 0000-0001-7607-0724 (2026). Critical one-arm probability for the metric Gaussian free field in low dimensions. Probability Theory and Related Fields, 195 (1-2). pp. 497-520. Springer Nature. ISSN 0178-8051

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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. Under the sole assumption that its sign clusters do not percolate, we derive an extension of Lupu’s formula for the two-point function at criticality. We then focus on the low-dimensional case 0 < ν < α 2 , where α governs the polynomial volume growth of G and ν the decay rate of the Green’s function on G. In particular, this includes the benchmark case G = Z3, for which α = 3 and ν = α − 2 = 1. We prove under these assumptions that the critical one-arm probability decays with distance R like R− ν 2 , up to multiplicative constants.

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 https://orcid.org/0000-0001-7607-0724 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-806739
Identification Number:	10.1007/s00440-025-01392-7
Journal or Publication Title:	Probability Theory and Related Fields
Volume:	195
Number:	1-2
Page Range:	pp. 497-520
Number of Pages:	24
Date:	15 June 2026
Publisher:	Springer Nature
ISSN:	0178-8051
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
Uncontrolled Keywords:	Keywords Language Percolation ; Gaussian free field ; Metric graph ; Critical one-arm probability English
['eprint_fieldname_oa_funders' not defined]:	Publikationsfonds UzK
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/80673