7 problems

Intermediate-dimension interpolation conjecture. There exists a monotone function $\alpha(s)\colon[0,1]\to[0,\infty)$ such that

Let $F$ be the limit set of Mandelbrot percolation with subdivision parameter $N\geq 2$ in dimension $d\geq 1$. Let $p_d$ denote the threshold for the dust phase and $p_c$ the crit…

Let $p\in(0,1)$ be the retention parameter in fractal percolation, and let property denote the strong-disconnection property defined in the source. Strong-disconnection conjecture.…

Let $A_\infty$ denote the final retained set in planar fractal percolation, and let $p_c$ be its critical parameter. Critical fractal-percolation carpet conjecture. At the critical…

Let $Q_d(p;m)$ and $\widehat Q_d(q;n)$ be the random sets appearing above, and let $A$ be a nonempty target set. In the critical case $2^dpq=1$, the relevant hitting probability is…

Let $N\geq 2$ and $d\geq 2$, and let $\mathcal C$ be the limiting set of Mandelbrot fractal percolation in $[0,1]^d$ with retention parameter $p$. Define the critical value … where…

Let $d\geq 2$ and $N\geq 2$. In full-space fractal percolation, let $\hat{p}_c(N,d)$ denote the threshold at which the limiting retained set ceases to be totally disconnected, and…