We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by Černý and Teixeira study by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.

中文翻译：

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在随机图，Wright常数和具有规定度的临界随机图上随机游走所留下的空集的几何

我们提供了一个显式算法，用于以给定的度数序列采样统一的简单连接随机图。该中心结果的副产品包括：（1）在一定规则性条件下，具有给定度数序列的统一简单连接图的连续标度极限和渐近性，具有给定度数序列的简单连接图的数目，无渐近性构造模型和均匀简单随机图模型的临界状态下，在次数序列有限的第三矩假设下，最大分量的空间结构。作为一种实质性的应用，我们通过获得随机正则图上的随机游走所留下的空置集中的最大分量的度量空间缩放极限，来回答Černý和Teixeira研究提出的问题。