5 problems

Let $\lpha_c$, $\lpha_u$, $\lpha_{\rm tr}$, and $\lpha_{\mathop{\rm b2b}}$ denote the thresholds defined in the paper for, respectively, percolation, uniqueness, truncated-model pe…

Fix $c\in(0,1)$. Let $\mathcal L_D^c$ be a Brownian loop soup of central charge $c$, let $\mathcal C$ be one of its clusters, and let $:\mathtt h_c\Theta_{\mathcal L_D^c}^{n-1}:$ d…

For $d\geq4$, consider the Brownian loop-soup in $(0,1)^d$ and the operation of contracting each loop, so that percolation can be viewed as a mechanism for gluing loops into cluste…

Let $d\in\{4,5\}$, and consider the cable-graph clusters in $(0,1)^d\cap\delta\mathbb Z^d$ as $\delta\to0$. Four- and five-dimensional scaling-limit conjecture. The limit in distri…

Let $d=3$, and let the cable-graph loop-soup clusters be formed on finer and finer meshes. Denote by $\Delta$ the Hausdorff dimension of the Brownian loop-soup clusters. Three-dime…