3 problems

Let $d=4$, let $m(\ell)\asymp\ell^{-\beta}$ be the worm-length mass function, and let $v_c$ denote the critical intensity. Four-dimensional worm-percolation conjecture. One has … T…

Let $d=3$, let $m(\ell)\asymp\ell^{-\beta}$ be the worm-length mass function, and let $v_c$ denote the critical intensity. Three-dimensional worm-percolation conjecture. One has ……

Let $\mathcal{L}$ be the random worm length and let $\mathrm{R}_*\asymp\sqrt{\mathcal{L}}$ denote the corresponding heuristic radius in the planar Poisson Boolean model. Write…