The ray nonsingularity of certain uniformly random ray patterns
The ray nonsingularity of certain uniformly random ray patterns
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DOI：
10.1016/j.laa.2017.10.003
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发表时间：
2018-01
影响因子：
1.1
通讯作者：
Yue Liu
中科院分区：
文献类型：
--
作者：
Yue Liu
具有给定的零-非零模式的均匀随机射线模式矩阵A（由没有多弧或环的有向图D描述）是其非零元素是均匀分布在复平面中的单位圆S1上的相互独立的随机变量的矩阵。本文证明了如果CG（D）是树，则I-A是射线非奇异的概率完全由D的圈图CG（D）（即D中有向圈的邻接结构）决定.给出了CG（D）是树时的概率计算公式，并证明了当CG（D）的阶趋于无穷大时，概率的极限为0。
A uniformly random ray pattern matrix A with a given zero-nonzero pattern (described by a digraph D with no multi-arcs or loops) is the matrix whose nonzero entries are mutually independent random variables uniformly distributed over the unit circle S 1 in the complex plane. It is shown in this paper that the probability of I− A to be ray nonsingular is completely determined by the cycle graph CG (D) of D (ie the adjacency structure of the directed cycles in D) if CG (D) is a tree. A formula is given to compute the probability when CG (D) is a tree, and it is also shown that as the order of CG (D) tends to infinity, the limit of the probability is 0.