The homology torsion growth of determinantal hypertrees

TL;DR

This paper studies the asymptotic behavior of the homology torsion growth in random determinantal hypertrees, establishing convergence of a normalized logarithmic torsion measure to a constant within bounds.

Contribution

It proves the convergence in probability of the normalized homology torsion growth in random determinantal hypertrees and bounds the limiting constant.

Findings

The normalized log of the homology torsion converges in probability.

The limiting constant c_d is bounded between (1/2) log((d+1)/e) and (1/2) log(d+1).

The result applies to random d-dimensional determinantal hypertrees.

Abstract

Fix a dimension $d\ge 2$, and let $T_n$ be a random $d$-dimensional determinantal hypertree on $n$ vertices. We prove that \[\frac{\log|H_{d-1}(T_n,\mathbb{Z})|}{{{n\choose {d}}}}\] converges in probability to a constant $c_d$, which satisfies \[\frac{1}2 \log\left(\frac{d+1}e\right)\le c_d\le \frac{1}2 \log\left(d+1\right) .\]