In *Perplexing Problems in Probability*, the following statement about first passage percolation (FPP) on $\mathbb{Z}^{d}$ is made. See e.g. this paper of Benjamini, Kalai and Schramm, where they quote from the section on FPP in the works of Kesten...

Novice readers might expect to hear next of a central limit theorem being proved,” writes Durrett, describing Kesten’s results, “however physicists tell us...that in two dimensions the standard deviation...is of order $|\mathbf{v}|^{1/3}$"

They are saying that geodesics on the lattice from the origin to a point $\mathbf{v}$ have a length with variance which converges to $C|\mathbf{v}|^{2/3}$ as $|\mathbf{v}| \to \infty$

But, a central limit theorem does apply. Do the novices simply miss the fact that the variance is non-trivial? Is there no such thing as a central limit theorem here?

Why would you *not* expect a central limit theorem to be proved for the geodesic length distribution, simply due to this sublinear variance?