  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.

Probability