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We further investigate properties of the Gaussian free field (GFF) on themetric graph associated to a discrete weighted graph (where the edges of thelatter are replaced by continuous line-segments of appropriate length) that hasbeen introduced by the first author. On such a metric graph, the GFF is arandom continuous function that generalises one-dimensional Brownian bridges sothat one-dimensional techniques can be used.  In the present paper, we define and study the pseudo-metric defined on themetric graph (and therefore also on the discrete graph itself), where thelength of a path on the metric graph is defined to be the local time at levelzero accumulated by the Gaussian free field along this path.  We first derive a pathwise transformation that relates the GFF on the metricgraph with the reflected GFF on the metric graph via the pseudo-distancedefined by the latter. This is a generalisation of Paul L\'evy's resultrelating the local time at zero of Brownian motion to the supremum of anotherBrownian motion. We also compute explicitly the distribution of certainfunctionals of this pseudo-metric and of the GFF. In particular, we point outthat when the boundary consists of just two points, the law of thepseudo-distance between them depends solely on the resistance of the networkbetween them.  We then discuss questions related to the scaling limit of this pseudo-metricin the two-dimensional case, which should be the conformally invariant way tomeasure distances between CLE(4) loops introduced and studied by the secondauthor with Wu, and by Sheffield, Watson and Wu. Our explicit laws on metricgraphs also lead to new conjectures for related functionals of the continuumGFF on fairly general Riemann surfaces.

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