Universality of the time constant for 2D critical first-passage percolation
Universality of the time constant for 2D critical first-passage percolation
复制标题
DOI：
10.1214/22-aap1808
复制
发表时间：
2019-04
期刊：
影响因子：
--
通讯作者：
M. Damron;Jack Hanson;Wai-Kit Lam
中科院分区：
文献类型：
--
作者：
M. Damron;Jack Hanson;Wai-Kit Lam
考虑点权为$(Tv)$的三角格子上的首次通过渗流问题，其公共分布函数$F$满足$F(0)=1/2$。这被称为FPP的关键情况，因为大的(关键的)零权重星团允许在距离为次线性的距离时间点之间传播。用$T(0，部分B(N))$表示从$0$到$x：xinty=n$的首次通过时间，我们证明了“时间常数”的存在，并找到了它的精确值为\[\lim_{n\to\inty}\frac{T(0，\artiB(N))}{\log n}=\frac{i}{2\sqrt{3}\pi}\Text{几乎肯定}，其中$i=\inf\{x>；0：F(X)>；1/2$和$F$是$t_v$的任何临界分布。这一结果表明，时间常数具有普适性，并且仅依赖于$i$的值。在最优矩条件下，我们得到了极限归一化方差的精确值，它也只是$I的函数。在假设时间常数存在的情况下，证明方法在其他二维晶格上也具有类似的普适性。
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 the "time constant'' and find its exact value to be \[ \lim_{n \to \infty} \frac{T(0,\partial B(n))}{\log n} = \frac{I}{2\sqrt{3}\pi} \text{ almost surely}, \] where $I = \inf\{x > 0 : F(x) > 1/2\}$ and $F$ is any critical distribution for $t_v$. This result shows that 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.