3 problems

Let $\mathfrak{p}\in(\mathfrak{p}_c(d),1)$, condition the origin to lie in the infinite cluster $\mathscr{C}_{\infty}$, and let $\overline{s}_n$ be the rescaled sandpiles defined f…

Let $\mathfrak{p}\in(\mathfrak{p}_c(d),1)$, and let $\Gamma_{\mathscr{C}_{\infty}}$ denote the set of symmetric $d\times d$ matrices admitting an integer-valued superharmonic repre…

Let $\mathfrak{p}\in(\mathfrak{p}_c(d),1)$, condition the origin to lie in the infinite cluster $\mathscr{C}_{\infty}$, and let $s_n$ be the stabilized sandpile obtained from $n$ c…