Universality of the time constant for $2D$ critical first-passage percolation

We consider first-passage percolation (FPP) on the triangular lattice with vertex weights $(t_v)$ whose common distribution function $F$ satisfies $F(0)=1\mathrm{/}2$. This is known as the critical case of FPP because large (critical) zero-weight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by $T(0,\mathrm{\partial }B(n))$ the first-passage time from $0$ to $\{x:\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}=n\}$, we show existence of the "time constant’’ and find its exact value to be $$\underset{n\to \mathrm{\infty }}{\lim }\frac{T(0,\mathrm{\partial }B(n))}{\log n}=\frac{I}{2\sqrt{3}\pi }\text{ almost surely},$$ where $I=\inf \{x>0:F(x)>1\mathrm{/}2\}$ and $F$ is any critical distribution for $t_v$. This result shows that the time constant is universal and depends only on the value of $I$. Furthermore, we find the exact value of the limiting normalized variance, which is also only a function of $I$, under the optimal moment condition on $F$. The proof method also shows an analogous universality on other two-dimensional lattices, assuming the time constant exists.