Chemical Distance for the Level Sets of the Gaussian Free Field

We consider the Gaussian free field $\phi$ on ${\mathbb{Z}}^d$ for $d\ge 3$ and study the level sets $\{\phi \ge h\}$ in the percolating regime. We prove upper and lower bounds for the probability that the chemical distance is much larger than Euclidean distance. Our proof uses a renormalization scheme combined with a bootstrap argument.