  We initiate the study on chemical distances of percolation clusters for level
sets of two-dimensional discrete Gaussian free fields as well as loop clusters
generated by two-dimensional random walk loop soups. One of our results states
that the chemical distance between two macroscopic annuli away from the
boundary for the random walk loop soup at the critical intensity is of
dimension 1 with positive probability. Our proof method is based on an
interesting combination of a theorem of Makarov, isomorphism theory and an
entropic repulsion estimate for Gaussian free fields in the presence of a hard
wall.
