We consider first-passage percolation on the two-dimensional triangular lattice $\mathcal{ T}$. Each site $v\in\mathcal{ T}$ 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 $c_n^+$ the first-passage time in $B^+(0,n)$ from $0$ to the half-circular boundary of $B^+(0,n)$. We prove \[\lim_{ n\rightarrow\infty}\frac{ c_n^+}{ \log n}=\frac{ \sqrt{ 3}}{ 2\pi}~ a.s.,~\lim_{ n\rightarrow\infty}\frac{ E c_n^+}{ \log n}=\frac{ \sqrt{ 3}}{ 2\pi},~\lim_{ n\rightarrow\infty}\frac{ \mathrm{ Var}(c_n^+)}{ \log n}=\frac{ 2\sqrt{ 3}}{ \pi}-\frac{ 9}{ \pi^2}.\] 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.
Categories: math.PR
