Hitting-time conjecture for the giant shadow and homological giant

Let $\mathcal{Y}_d(n)$ be the stochastic Linial–Meshulam process in dimension $d\geq2$. Let $LT(n)$ denote its largest torsion group, and for a $d$-complex $Y$ with complete $(d-1)$-skeleton over $\mathbb{R}$ let $SH_{\mathbb{R}}(Y)$ be its shadow. A core is a subcomplex in which every $(d-1)$-face lies in at least two $d$-faces. Hitting-time conjecture. There exists a constant $\delta$ depending on $d$ such that, with high probability, there is an $m_0\in\{1,\ldots,\binom{n}{d+1}\}$ for which: (1) the torsion part of $H_{d-1}(Y_d(n,m_0))$ is isomorphic to $LT(n)$; (2) $Y_d(n,m_0)$ contains a spanning core $Y'$ with $H_{d-1}(Y')\cong LT(n)$, whereas $Y_d(n,m_0-1)$ contains no such core; and (3) $|SH_{\mathbb{R}}(Y_d(n,m_0))|\geq\delta n^{d+1}$ while $|SH_{\mathbb{R}}(Y_d(n,m_0-1))|\leq O(n)$. This conjecture identifies the hitting time of the homological giant with that of the giant shadow and predicts a one-step jump from a small to a giant shadow; the source attributes the shadow-jump prediction to Linial and Peled and presents the stronger structural statement as conjectural.