We introduce a model for random geodesic drawings of the complete bipartite graph $K_{n,n}$ on the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$, where we select the vertices in each bipartite class of $K_{n,n}$ with respect to two non-degenerate probability measures on $\mathbb{S}^2$. It has been proved recently that many such measures give drawings whose crossing number approximates the Zarankiewicz number (the conjectured crossing number of $K_{n,n}$). In this paper we consider the intersection graphs associated with such random drawings. We prove that for any probability measures, the resulting random intersection graphs form a convergent graph sequence in the sense of graph limits. The edge density of the limiting graphon turns out to be independent of the two measures as long as they are antipodally symmetric. However, it is shown that the triangle densities behave differently. We examine a specific random model, blow-ups of antipodal drawings $D$ of $K_{4,4}$, and show that the triangle density in the corresponding crossing graphon depends on the angles between the great circles containing the edges in $D$ and can attain any value in the interval $\bigl(\frac{83}{12288}, \frac{128}{12288}\bigr)$.

中文翻译：

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限制球体上测地线图的交叉数

我们在 $\mathbb{R}^3$ 中的单位球体 $\mathbb{S}^2$ 上引入了完整二部图 $K_{n,n}$ 的随机测地线绘图模型，我们选择顶点在 $K_{n,n}$ 的每个二分类中，关于 $\mathbb{S}^2$ 上的两个非退化概率度量。最近已经证明，许多这样的度量给出的图形的交叉数接近 Zarankiewicz 数（$K_{n,n}$ 的推测交叉数）。在本文中，我们考虑与此类随机绘图相关的交集图。我们证明，对于任何概率度量，产生的随机交叉图在图限制的意义上形成收敛图序列。只要它们是对映对称的，极限图形的边缘密度就与这两个度量无关。然而，结果表明，三角形密度的行为不同。我们检查了一个特定的随机模型，放大了 $K_{4,4}$ 的对映图 $D$，并表明相应交叉图形中的三角形密度取决于包含 $K_{4,4}$ 中边的大圆之间的角度D$ 并且可以在区间 $\bigl(\frac{83}{12288}, \frac{128}{12288}\bigr)$ 中获得任何值。