$\mathbb{R}^d$上的不变二部随机图
Invariant bipartite random graphs on $\mathbb{R}^d$
摘要：
假设$\mathbb{R}^d$中的红点和蓝点分别按照两个有限强度的简单点过程$λ_{\mathcal{R}}$和$λ_{\mathcal{B}}$出现。此外，设$ν$和$μ$是严格正整数上的两个概率分布。分别使用规则$ν$和$μ$为每个红色和蓝色点独立分配随机数量的存根（半边）。我们感兴趣的是，以获得随机二部图，其中每个点有一个规定的度分布与法律$ν$或$μ$取决于它的颜色之间的匹配存根的消隐不变的计划。令$X$和$Y$分别为具有规律$ν$和$μ$的随机变量。对于一个大类的点过程，我们表明，我们可以得到这样的竞争不变的计划匹配a.s.所有存根当且仅当\[ λ_{\mathcal{R}} \mathbb{E}(X)= λ_{\mathcal{B}} \mathbb{E}(Y), \]允许$\infty$在两侧，当两个法律有无限的意思。此外，我们还研究了一个基于Gale-Shapley稳定婚姻的特殊方案。对于这个方案，我们给出了关于$X$和$Y$存在和不存在无穷分支的充分条件.这些结果是由Deijfen，Häggström和Holroyd获得的双色版本。
英文摘要：
Suppose that red and blue points occur in $\mathbb{R}^d$ according to two simple point process with finite intensities $λ_{\mathcal{R}}$ and $λ_{\mathcal{B}}$, respectively. Furthermore, let $ν$ and $μ$ be two probability distributions on the strictly positive integers. Assign independently a random number of stubs (half-edges) to each red and blue point with laws $ν$ and $μ$, respectively. We are interested in translation-invariant schemes to match stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law $ν$ or $μ$ depending on its color. Let $X$ and $Y$ be random variables with law $ν$ and $μ$, respectively. For a large class of point processes we show that we can obtain such translation-invariant schemes matching a.s. all stubs if and only if \[ λ_{\mathcal{R}} \mathbb{E}(X)= λ_{\mathcal{B}} \mathbb{E}(Y), \] allowing $\infty$ in both sides, when both laws have infinite mean. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage. For this scheme we give sufficient conditions on $X$ and $Y$ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Häggström and Holroyd.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
