Chemical Distance for the Level Sets of the Gaussian Free Field

Abstract: We consider the Gaussian free field $\varphi$ on $\mathbb{Z}<sup>d$ for $d \geq 3$ and study the level sets ${\varphi \geq h }$ 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.