The homological connectivity threshold conjecture for k-fold coverage

Let $B_r^{(k)}$ be the $k$-fold coverage complex on the $d$-dimensional torus, let ${\cal{H}}_{i,r}^{(k)}$ be the event that $H_i(B_s^{(k)})\cong H_i({\mathbb{T}}^d)$ for every $s\geq r$, and let $\Lambda$ be the model's intensity parameter. Let $k\geq 1$ and $0\leq i\leq d-2$, and suppose that $w(n)\rightarrow\infty$ as $n\rightarrow\infty$.

Homological connectivity threshold conjecture.

$$\lim_{n\rightarrow\infty}\mathbb{P}({\cal{H}}_{i,r}^{(k)}) = \begin{cases} 1 & \Lambda = \log n + (i+k-2)\log\log n + w(n), \\ 0 & \Lambda = \log n + (i+k-2)\log\log n - w(n). \end{cases}$$

The preceding corollary gives only an upper bound for these homological connectivity thresholds, while the conjecture is motivated by the homological connectivity results of Bobrowski (2022).