5 problems

Let $cP(cell^2_\downarrow)$ be the space of probability measures on $cell^2_\downarrow$, and let $\mathscr R=\mathscr R_{d,L,\alpha}$ be the renormalization map obtained by indepen…

At the critical dimension $d=3\alpha$, compare hierarchical percolation on $cLambda_n$ with free and periodic boundary conditions, where $L$ is the hierarchical scale parameter. Pe…

For fixed $p$ and $k$, consider the random norm $cullX_{n,t}cul_p^p$ and its variance. Central-limit conjecture. … These are the moments of a standard normal distribution, so the c…

At the upper-critical dimension $d=3\alpha$, let $M_n$ denote the typical size of the largest cluster and let $L$ be the hierarchical scale parameter. Largest-cluster conjecture. F…

Consider critical hierarchical percolation with size-biased cluster $K_n$, viewed as a metric measure space after suitable rescaling. Continuum-random-tree conjecture. The size-bia…