Abstract
We prove that first-passage percolation times across thin cylinders of the form [0, n] × [−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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Chatterjee, S., Dey, P.S. Central limit theorem for first-passage percolation time across thin cylinders. Probab. Theory Relat. Fields 156, 613–663 (2013). https://doi.org/10.1007/s00440-012-0438-z
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DOI: https://doi.org/10.1007/s00440-012-0438-z