4 problems

Let $n=2$ or $n=3$, let $V_N$ denote the finite domain of the membrane model, let $\psi=(\psi_x)_{x\in V_N}$ be its field, let $\operatorname{E}_N$ denote expectation, and let…

Let $v>\gamma^{\circ}$ and let $\mathbbm{Q}^{U_v}=\lim_{a\to\infty}\mathbbm{W}(\mathord\cdot\mid\mathcal{E}_a^{U_v})$ be any measure supplied by the critical-speed framework. Let…

Let $\gamma^{\circ}$ and $\gamma^{\bullet}$ be the critical speeds with $1<\gamma^{\circ}<\gamma^{\bullet}$, and let $\mathcal{E}_a^{U_v}$ be the conditioning events associated wit…

Let $U\subseteq\mathbbm{R}\times[0,\infty)$ be open with $(0,0)\in U$, and let $\operatorname{cost}(P)$ be the path cost. Assume that there is a unique path $P$ from $(0,0)$) to…