# Central limit theorem is expected, but variance is apparently sublinear? So?

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?

• Ok I will move it there
– apkg
Feb 12 '19 at 14:59
• Actually, it fits here. The short answer to the question is that by the statement, Durrett really meant "CLT with variance $n^{1/2}$", and the exponent $1/3$ already tells you that something unusual is going on. In fact, one does not expect a Gaussian limit (as in "CLT") but rather a Tracy-Widom limit. In the case of exponential weights, a version of this result is proved by Johansson (Theorem 1.2 in arxiv.org/pdf/math/9903134.pdf). See also arxiv.org/pdf/math/9910146.pdf for a related result and arxiv.org/pdf/1802.00729.pdf for a recent perspective and references. Feb 12 '19 at 18:33
• I see - my apologies to the OP for a hasty reading. Feb 12 '19 at 21:17
• Thank you all. So, the travel time is Tracy-Widom. What about the transversal deviations of the geodesics? Is the distribution also Tracy-Widom?
– apkg
Feb 22 '19 at 12:06