The ray nonsingularity of certain uniformly random ray patterns

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 S1 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 (i.e. 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.