The ant on loops: Alexander-Orbach conjecture for the critical level set of the Gaussian free field

Alexander and Orbach (AO) in 1982 conjectured that the simple random walk on critical percolation clusters (also known as the ant in the labyrinth) in Euclidean lattices exhibit mean field behavior; for instance, its spectral dimension is $4/3$. While false in low dimensions, this is expected to be true above the upper critical dimension of six. First rigorous results in this direction go back to Kesten who verified this on the tree. After many developments, in a breakthrough work, Kozma and Nachmias [KN] established the AO conjecture for bond percolation on $\mathbb{Z}^d$ for $d>19$ and $d>6$ for the spread out lattice. We investigate the validity of the AO conjecture for the critical level set of the Gaussian Free Field (GFF), a canonical dependent percolation model of central importance. In an influential work, Lupu proved that for the cable graph of $\mathbb{Z}^d$ (which is obtained by also including the edges), the signed clusters of the associated GFF are given by the corresponding clusters in a Poisson loop soup, thus reducing the analysis to the study of the latter. In 2021, Werner put forth an evocative picture for the critical behavior in this setting drawing an analogy with the usual bond case. Building on this, Cai and Ding established the universality of the extrinsic one arm exponent for all $d > 6.$ In this article, we carry this program further, and consider the random walk on sub-sequential limits of the cluster of the origin conditioned to contain far away points. These form candidates for the Incipient Infinite Cluster (IIC), first introduced by Kesten in the planar case. Inspired by the program of [KN], introducing several novel ideas to tackle the long range nature of this model and its effect on the intrinsic geometry of the percolation cluster, we establish that the AO conjecture indeed holds for any sub-sequential IIC, for all large enough dimensions.