Critical first-passage percolation starting on the boundary

Abstract

We consider first-passage percolation on the two-dimensional triangular lattice T. Each site v∈T is assigned independently a passage time of either 0 or 1 with probability 1∕2. Denote by B+(0,n) the upper half-disk with radius n centered at 0, and by cn+ the first-passage time in B+(0,n) from 0 to the half-circular boundary of B+(0,n). We prove limn→∞cn+logn=32πa.s.,limn→∞Ecn+logn=32π,limn→∞Var(cn+)logn=23π−9π2.These results enable us to prove limit theorems with explicit constants for any first-passage time between boundary points of Jordan domains. In particular, we find the explicit limit theorems for the cylinder point to point and cylinder point to line first-passage times.