Central limit theorem for firstpassage percolation time across thin cylinders
Abstract
We prove that firstpassage percolation times across thin cylinders of the form $[0,n]\times [h_n,h_n]^{d1}$ 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2009
 DOI:
 10.48550/arXiv.0911.5702
 arXiv:
 arXiv:0911.5702
 Bibcode:
 2009arXiv0911.5702C
 Keywords:

 Mathematics  Probability;
 60F05;
 60K35
 EPrint:
 Final version, accepted in Probability Theory and Related Fields. 40 pages, 7 figures