4 problems

Let $\mathcal{L}_{1/2}^{\widetilde{\mathbb{Z}}^3}$ be the loop soup on the metric graph of $\mathbb{Z}^3$, and let $\mathcal{L}_{1/2}^{\mathbb{R}^3}$ be the corresponding loop soup…

For functions $f$ and $g$ depending on the dimension $d$, define … Here a critical cluster is either a loop cluster of $\widetilde{\mathcal{L}}_{1/2}$ or, when non-empty, a positiv…

Let $\theta \in (0,1/2)$, let $h_{\theta,N}$ be the discrete loop-soup field, and define its Wick powers by … where $G(x,x)$ is the relevant Green function and…

Loop-cluster metric conjecture. There is a metric on the clusters of $\mathcal{L}_{g,1/2}$ inside $\Sigma$, measurable with respect to $\mathcal{L}_{g,1/2}$, obtained as the limit…