4 problems

Let $c_n^{\mathbb{T}}$ be the $n$-step weakly self-avoiding-walk count on the torus, let $\mu$ be the growth constant on $\mathbb{Z}^d$, and let $V$ be the number of vertices in th…

Let $V$ be the number of vertices in a $d$-dimensional torus, let $z_c$ be the critical point, and let $L$ denote the length under the variable-length weakly self-avoiding-walk mea…

Let $\mathbb T_{r,d}$ be a high-dimensional torus with $d>6$, let $V=r^d$ be its number of vertices, and write $|\mathcal C_{(1)}|\ge |\mathcal C_{(2)}|\ge\cdots$ for the ordered s…

Let $\mathbb T_{r,d}$ be either the nearest-neighbor or spread-out torus described above, with $d>6$, and let $V=r^d$ be its number of vertices. Consider bond percolation on…