We consider 2‐dimensional random simplicial complexes Y in the multiparameter model. We establish the multiparameter threshold for the property that every 2‐dimensional simplicial complex S admits a topological embedding into Y asymptotically almost surely. Namely, if in the procedure of the multiparameter model on n vertices, each i‐dimensional simplex is taken with probability p_i = p_i(n), then the threshold is . Our claim in one direction is in fact slightly stronger, namely, we show that if is sufficiently larger than then every S has a fixed subdivision S′ which admits a simplicial embedding into Y asymptotically almost surely. In the other direction we show that if is sufficiently smaller than , then asymptotically almost surely, the torus does not admit a topological embedding into Y.

中文翻译：

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拓扑嵌入到随机2复合体中

我们在多参数模型中考虑了二维随机单纯形复数Y。我们为几乎确定每个二维简单复形S渐近地将拓扑嵌入Y的性质建立了多参数阈值。即，如果在n个顶点的多参数模型的过程中，每个i维单形以概率p _i  =  p _i（n）进行，则阈值为。实际上，我们在一个方向上的主张稍微强一些，也就是说，我们表明，如果大于，则每个S具有固定的细分S'，几乎可以肯定地渐近地将单纯形嵌入到Y中。在另一个方向上，我们表明如果小于，则几乎可以肯定地渐近地，圆环不允许将拓扑嵌入到Y中。