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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
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References:

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