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The $2$-torsion of determinantal hypertrees is not Cohen-Lenstra
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abstract
Let $T_n$ be a $2$-dimensional determinantal hypertree on $n$ vertices. Kahle and Newman conjectured that the $p$-torsion of $H_1(T_n,\mathbb{Z})$ asymptotically follows the Cohen-Lenstra distribution. For $p=2$, we disprove this conjecture by showing that given a positive integer $h$, for all large enough $n$, we have \[\mathbb{P}(\dim H_1(T_n,\mathbb{F}_2)\ge h)\ge \frac{e^{-200h}}{(100h)^{5h}}.\] We also show that $T_n$ is a bad cosystolic expander with positive probability.
Forward citations
Cited by 2 Pith papers
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Local limits of determinantal processes
The local limit of the determinantal process on a C4-free (d,k+1)-bi-regular bipartite graph, as d tends to infinity, is a multi-type Poisson(k) branching tree T_k.
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Using dense graph limit theory to count cocycles of random simplicial complexes
For every prime p, the dimension of H_1(T_n, F_p) of a random 2-dimensional determinantal hypertree and of a random 1-out 2-complex is o(n^2) in probability.
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