AI 中文总结
该研究针对随机二维行列式超树的上同调，刻画了小支撑集对应的有限连通图，明确了素数p>2和p=2时的不同结论，为相关拓扑问题提供了理论结果。
AI 中文摘要
设$T_n$为$n$个顶点上的随机二维行列式超树。给定任意素数$p$，我们回答如下问题：若$Z^1(T_n,\boldsymbol{F}_p)$中的一个上同调具有小支撑集，该支撑集通常是什么样的？更确切地说，我们刻画了所有有限连通图$G$，使得存在常数$c_G>0$，满足如下性质：对所有足够大的$n$，以至少$c_G$的概率存在上同调$f \in Z^1(T_n,\boldsymbol{F}_p)$，使得移除所有孤立顶点后，$f$的支撑集与$G$同构。我们证明，当$p>2$时，不存在任何此类图；当$p=2$时，一个连通图具有上述性质当且仅当它有唯一的奇长度环，且若该唯一环为三角形，则要求三角形的所有顶点度数至少为3。
英文摘要
Let ${T}_n$ be a random $2$-dimensional determinantal hypertree on $n$ vertices. Given any prime $p$, we answer the following question: If a cocycle in $Z^1({T}_n,\mathbb{F}_p)$ has small support, what does the support typically look like? More precisely, we characterize all the finite connected graphs $G$ for which there is a constant $c_G>0$ with the following property: For all large enough $n$, with probability at least $c_G$, we have a cocycle $f\in Z^1({T}_n,\mathbb{F}_p)$ such that after removing all the isolated vertices, the support of $f$ is isomorphic to $G$. We prove that for $p>2$, we do not have any such graph. For $p=2$, a connected graph has the property above if and only if it has a unique cycle such that this unique cycle has odd length, moreover, if the unique cycle is a triangle, then we also need to require that all the vertices of the triangle have degree at least $3$.
