Critical first-passage percolation starting on the boundary

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 $1\mathrm{/}2$. 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 $$\underset{n\to \mathrm{\infty }}{\lim }\frac{c_n^{+}}{\log n}=\frac{\sqrt{3}}{2\pi }a\mathrm{.}s\mathrm{.},\underset{n\to \mathrm{\infty }}{\lim }\frac{E{c}_n^{+}}{\log n}=\frac{\sqrt{3}}{2\pi },\underset{n\to \mathrm{\infty }}{\lim }\frac{\mathrm{V}\mathrm{a}\mathrm{r}(c_n^{+})}{\log n}=\frac{2\sqrt{3}}{\pi }-\frac{9}{{\pi }^2}\mathrm{.}$$ 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.