7 problems

Let $\alpha_c^{(k)}(d)$ denote the critical value of the parameter $\alpha$ for percolation in the GN$_k(d,\alpha)$ model in dimension $d$, with integer $k\geq 1$. High-dimensional…

Let $p_c(d,R)$ denote the critical probability for the existence of an infinite connected cluster in the directed space-time percolation model with spatial range $R$, in dimension…

Let $a,b,c,r$ be parameters for the family of neighbourhoods $\mathcal N_{r}^{a,b,c}$, let $L_c(\mathcal N_{r}^{a,b,c},p)$ denote its critical length, and define $\alpha_1=1/2$. Fo…

For $d\geq 3$, let $p_{1,c}(\mathbb{Z}^d)$ denote the critical probability for $1$-independent percolation on the integer lattice. High-dimensional percolation conjecture. There ex…

Let $Q_n$ be the $n$-dimensional hypercube. The critical probability $p_{1,c}(\mathbb{Z}^2\times Q_n)$ is defined for $1$-independent percolation on this product graph. Hypercube p…

Let $p_c^{(\mathrm{M})}([n]^d)$ denote the critical probability for percolation in the modified bootstrap process on $[n]^d$, namely the infimum over $p$ such that the probability…

Let $p_c([n]^2,2)$ denote the critical probability for percolation in the two-neighbour bootstrap process on the square grid $[n]^2$. Gravner and Holroyd proved the upper bound … f…