Transitions for exceptional times in dynamical first-passage percolation
Transitions for exceptional times in dynamical first-passage percolation
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DOI：
10.1007/s00440-022-01178-1
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发表时间：
2021-08
影响因子：
2
通讯作者：
M. Damron;Jack Hanson;David Harper;Wai-Kit Lam
中科院分区：
文献类型：
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作者：
M. Damron;Jack Hanson;David Harper;Wai-Kit Lam
在首次通过渗流(FPP)中，我们给出了图的顶点上的非负权，并研究了距离较远的顶点之间的最小路的权重。如果f是的分布函数，则有不同的区域：如果f(0)较小，该权重通常以距离的线性函数形式增长，而当f(0)较大时，权重通常为一阶。介于这两者之间的是一个关键的制度，在这个制度中，重量可能会出现差异，但这种差异是次要的。我们研究了三角格子上临界FPP的一个动态版本，其中顶点根据独立的1率Poisson过程重采样。我们证明，如果，那么A.S.存在体重非典型增长的特殊时间，但如果，那么A.S.不存在这样的时间。此外，在前一种情况下，我们计算了例外集的Hausdorff和Minkowski维，并证明了它们可以相等，但不必相等。这些结果显示了比人们在亚临界(通常)FPP中看到的更广泛的动力学行为。
In first-passage percolation (FPP), we letbe i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. IfFis the distribution function of, there are different regimes: ifF(0) is small, this weight typically grows like a linear function of the distance, and whenF(0) is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. We study a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We prove that if, then a.s. there are exceptional times at which the weight grows atypically, but if, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP.