Point-to-point distance in first passage percolation on (Tree)xZ

Abstract

We consider first passage percolation (FPP) on Tdx G, where Td is the d-regular tree (d ≥ 3)and G is a graph containing an infinite ray 0,1,2,….It is shown that for a fixed vertex v in the tree, the fluctuation of the distance in the FPP metric between the points (v, 0) and (v, n) is of the order of at most log n. We conjecture that the real fluctuations are of order 1 and explain why.