## Universality of the time constant for $$2D$$ critical first-passage percolation.(English)Zbl 1525.60110

Summary: We consider first-passage percolation (FPP) on the triangular lattice with vertex weights $$(t_v)$$ whose common distribution function $$F$$ satisfies $$F(0)=1/2$$. 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 $$T(0,\partial B(n))$$ the first-passage time from 0 to $$\{x:\| x\|_{\infty}=n\}$$, we show existence of a “time constant” and find its exact value to be $\lim\limits_{n\to\infty}\frac{T(0,\partial B(n))}{\log n}=\frac{I}{2\sqrt{3}\pi}\text{ almost surely},$ where $$I=\inf \{ x>0:F(x)> 1/2\}$$ and $$F$$ is any critical distribution for $$t_v$$. This result shows that this 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.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation

### Keywords:

first-passage percolation; time constant; universality
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### References:

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