Probability Theory and Related Fields

, Volume 171, Issue 3–4, pp 775–818 | Cite as

The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph

  • Titus LupuEmail author
  • Wendelin Werner


We further investigate properties of the Gaussian free field (GFF) on the metric graph associated to a discrete weighted graph (where the edges of the latter are replaced by continuous line-segments of appropriate length) that has been introduced by the first author. On such a metric graph, the GFF is a random continuous function that generalises one-dimensional Brownian bridges so that one-dimensional techniques can be used. In the present paper, we define and study the pseudo-metric defined on the metric graph (and therefore also on the discrete graph itself), where the length of a path on the metric graph is defined to be the local time at level zero accumulated by the Gaussian free field along this path. We first derive a pathwise transformation that relates the GFF on the metric graph with the reflected GFF on the metric graph via the pseudo-distance defined by the latter. This is a generalisation of Paul Lévy’s result relating the local time at zero of Brownian motion to the supremum of another Brownian motion. We also compute explicitly the distribution of certain functionals of this pseudo-metric and of the GFF. In particular, we point out that when the boundary consists of just two points, the law of the pseudo-distance between them depends solely on the resistance of the network between them. We then discuss questions related to the scaling limit of this pseudo-metric in the two-dimensional case, which should be the conformally invariant way to measure distances between CLE(4) loops introduced and studied by the second author with Wu, and by Sheffield, Watson and Wu. Our explicit laws on metric graphs also lead to new conjectures for related functionals of the continuum GFF on fairly general Riemann surfaces.


Random distances Gaussian free field Metric graph Local time Conformal loop ensemble 

Mathematics Subject Classification

60G60 60G15 60K35 60J55 60J65 60J67 



T.L. acknowledges the support of Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation. W.W. acknowledges the support of the SNF Grant SNF-155922. The authors are also part of the NCCR Swissmap of the SNF. They also thank the referees for their comments.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institute for Theoretical StudiesETH ZürichZurichSwitzerland
  2. 2.Department of MathematicsETH ZürichZurichSwitzerland

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