Central limit theorem for first-passage percolation time across thin cylinders

Details Central limit theorem for first-passage percolation time across thin cylinders Central limit theorem for first-passage percolation time across thin cylinders

2009-11-30

arxiv.org/abs/0911.5702v3

Mathematics

Probability

38 pages, 7 figures, Results on CLT upto the height of the optimal path added

Scientific paper

We prove that first-passage percolation times across thin cylinders of the form $[0,n]\times [-h_n,h_n]^{d-1}$ obey Gaussian central limit theorems as long as $h_n$ grows slower than $n^{1/(d+1)}$. It is an open question as to what is the fastest that $h_n$ can grow so that a Gaussian CLT still holds. Under the natural but unproven assumption about existence of fluctuation and transversal exponents, and strict convexity of the limiting shape in the direction of $(1,0,...,0)$, we prove that in dimensions 2 and 3 the CLT holds all the way up to the height of the unrestricted geodesic. We also provide some numerical evidence in support of the conjecture in dimension 2.

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