We consider limit probabilities of first order properties in random graphs with a given degree sequence. Under mild conditions on the degree sequence, we show that the closure set of limit probabilities is a finite union of closed intervals. Moreover, we characterize the degree sequences for which this closure set is the interval $[0,1]$, a property that is intimately related with the probability that the random graph is acyclic. As a side result, we compile a full description of the cycle distribution of random graphs and study their fragment (disjoint union of unicyclic components) in the subcritical regime. Finally, we amend the proof of the existence of limit probabilities for first order properties in random graphs with a given degree sequence; this result was already claimed by Lynch~[IEEE LICS 2003] but his proof contained some inaccuracies.

翻译：我们研究了具有给定度序列的随机图中一阶性质的极限概率。在度序列满足温和条件下，我们证明极限概率的闭包集是有限个闭区间的并集。此外，我们刻画了使该闭包集为区间$[0,1]$的度序列特征，这一性质与随机图为无环图的概率密切相关。作为附带结果，我们完整描述了随机图的环分布，并研究了其在亚临界状态下的碎片（即单环分支的不交并）。最后，我们修正了关于具有给定度序列的随机图中一阶性质极限概率存在性的证明；该结果已由Lynch~[IEEE LICS 2003]提出，但其原始证明存在若干不准确之处。