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?

  • $\begingroup$ Ok I will move it there $\endgroup$
    – apkg
    Feb 12 '19 at 14:59
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    $\begingroup$ 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. $\endgroup$ Feb 12 '19 at 18:33
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    $\begingroup$ I see - my apologies to the OP for a hasty reading. $\endgroup$ Feb 12 '19 at 21:17
  • $\begingroup$ Thank you all. So, the travel time is Tracy-Widom. What about the transversal deviations of the geodesics? Is the distribution also Tracy-Widom? $\endgroup$
    – apkg
    Feb 22 '19 at 12:06

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