5 problems

Consider critical percolation on a high-dimensional torus of volume $V$, with side length $r$, and let ${\cal C}_{\rm max}$ be a largest critical cluster. Largest-cluster scaling-l…

Consider critical percolation in a box of width $r$ in dimension $d>6$, and let $V=r^d$. Under free boundary conditions, clusters are connected only within the box; under periodic…

Let $d>6$ and consider critical percolation on a high-dimensional torus, with ${\cal C}_{\rm max}$ denoting a maximal critical cluster. Random graph asymptotics means that the maxi…

Aizenman's critical torus scaling conjecture. The largest cluster should have size of order

Let $5malpha(5mathbb Z^d)$ be the critical dynamical-percolation speed exponent for the lattice $5mathbb Z^d$, and let $5malpha(5mathbb T_{b+1})$ be the corresponding exponent for…