# Local time at zero metric associated to a GFF on a cable graph.

We will consider an electrical network and replace the discrete edges by continuous lines of appropriate length. This is a cable graph. The discrete Gaussian Free Field (GFF) on the network can be interpolated to a continuous process on the cable graph and which satisfies the Markov property. This is the cable GFF . We will consider a pseudo-metric on the cable graph related to the local time at zero of the cable GFF . We will compute some universal explicit laws for this metric and show a generalization on the cable graph of Lévy’s theorem for the local time at zero of a Brownian motion. We will conjecture that on an approximation of a simply connected planar domain our pseudo-metric converges to the conformal invariant metric on CLE4 loops given by the CLE4 growth process. Moreover, identities on the cable graph lead by convergence to some explicit distributions for some local sets of two-dimensional continuum GFF , not only on a simply connected domain of C but on general Riemann surfaces.