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Topological embeddings into random 2‐complexes
Random Structures and Algorithms ( IF 0.9 ) Pub Date : 2021-01-06 , DOI: 10.1002/rsa.20987
Michael Farber 1 , Tahl Nowik 2
Affiliation  

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 pi = pi(n), then the threshold is urn:x-wiley:rsa:media:rsa20987:rsa20987-math-0001. Our claim in one direction is in fact slightly stronger, namely, we show that if urn:x-wiley:rsa:media:rsa20987:rsa20987-math-0002 is sufficiently larger than urn:x-wiley:rsa:media:rsa20987:rsa20987-math-0003 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 urn:x-wiley:rsa:media:rsa20987:rsa20987-math-0004 is sufficiently smaller than urn:x-wiley:rsa:media:rsa20987:rsa20987-math-0005, then asymptotically almost surely, the torus does not admit a topological embedding into Y.

中文翻译:

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 in)进行,则阈值为骨灰盒:x-wiley:rsa:media:rsa20987:rsa20987-math-0001。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,如果骨灰盒:x-wiley:rsa:media:rsa20987:rsa20987-math-0002大于,骨灰盒:x-wiley:rsa:media:rsa20987:rsa20987-math-0003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,我们表明如果骨灰盒:x-wiley:rsa:media:rsa20987:rsa20987-math-0004小于骨灰盒:x-wiley:rsa:media:rsa20987:rsa20987-math-0005,则几乎可以肯定地渐近地,圆环不允许将拓扑嵌入到Y中
更新日期:2021-01-06
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