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 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 precise estimates of percolation probabilities in different regimes which altogether describe a sharp phase transition.
"Percolation for level-sets of Gaussian free fields on metric graphs." Ann. Probab. 48 (3) 1411 - 1435, May 2020. https://doi.org/10.1214/19-AOP1397