4 problems

Let $d\geq 3$, and let $t_k^{\mathrm{perc}}$ be the critical threshold for homological percolation in degree $k$, while $t_k^{\mathrm{ec}}$ denotes the corresponding zero of the ex…

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 $\lambda_c$ an…

Let $M$ be a nice compact space of maximal possible homological degree $d$, and let $t_k^{\mathrm{perc}}$ denote the critical threshold for the appearance of giant $k$-cycles in a…

Let $M$ be a nice compact space, let $\{X_t\}$ be a random filtration with $X_t\subset M$, and fix a homological degree $k$. For the inclusion $i:X_t\hookrightarrow M$, define … He…