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

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.