  We study level-set percolation for Gaussian free fields on metric graphs. In
two dimensions, we give an upper bound on the chemical distance between the two
boundaries of a macroscopic annulus. Our bound holds with high probability
conditioned on connectivity and the bound is sharp up to a poly-logarithmic
factor with an exponent of one-quarter. This substantially improves a previous
result by Li and the first author. In three dimensions and higher, we provide
rather sharp estimates of percolation probabilities in different regimes which
altogether describe a sharp phase transition.
