  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.
