Sharp and ordered phase transitions for giant homological cycles

Let $d$ be the dimension of the torus and let $\lambda_{0,k}$ and $\lambda_{1,k}$ denote the lower and upper thresholds for the appearance of giant $k$-cycles, while $\lambda_c$ and $\bar \lambda_c$ are the thresholds for the giant component and the final giant cycle, respectively. For $0<k<d$, define $\lambda_k:=\lambda_{0,k}=\lambda_{1,k}$. Sharp and ordered phase-transition conjecture. For all $0<k<d$ we have $\lambda_{0,k}=\lambda_{1,k}:=\lambda_k$, and, in addition,

$$\lambda_c=\lambda_1<\lambda_2<\cdots<\lambda_{d-1}=\bar\lambda_c.$$

The conjecture strengthens the theorem's bounds by asserting sharpness of every phase transition and strict ordering of the thresholds at which giant cycles of successive dimensions appear; the paper states that proving this remains future work.