Determine boundary-regime asymptotics for cycle-cover probability

Determine the precise asymptotic behavior of \(\Pr(\mathcal N)\), equivalently the probability that a directed graph sampled from a directed step-graphon admits cycle covers of all required sizes, when the concentration vector \(x^*\) lies on the boundary of the cycle polytope \(\vec{\mathcal X}(S)\) while satisfying feasibility but not strict feasibility.

Background

The main theorems characterize convergence of $\Pr(\mathcal N)$ to one away from the boundary of the cycle polytope. When $$x^*\in\vec{\mathcal X}(S)\setminus\operatorname{relint}\vec{\mathcal X}(S)$$, the paper proves only an upper bound of one-half on the limiting superior of the probability that the sampled graph has a cycle cover of all vertices. The exact asymptotics may depend on the face containing $x^*$, as in the previously studied symmetric case.