Łuczak–Peled conjecture on torsion in random Linial–Meshulam complexes

Let $Y_d(n,p)$ be the random $d$-dimensional Linial–Meshulam complex, let $H_{d-1}(Y_d(n,p))$ denote its $(d-1)$st homology group, and let $c_d$ be the constant appearing in the torsion-burst threshold. Here $p=p(n)$ may depend on $n$.

Łuczak–Peled conjecture. For every $d \geq 2$ and $p=p(n)$ such that $|np-c_d|$ is bounded away from $0$, $H_{d-1}(Y_d(n,p))$ is torsion-free asymptotically almost surely.

The conjecture asserts that torsion occurs only in a narrow window around the critical value $np=c_d$, so the torsion burst has a unique phase. The source describes proving the existence of the burst as difficult and presents uniqueness as the more tractable open problem.