12 problems

For $d\in\\{4,5\\}$, consider the loop soups $\mathcal{L}_{1/2}^{\mathbb{R}^d}$ and $\mathcal{L}_{1/2}^{\widetilde{\mathbb{Z}}^d}$. In dimensions $d=4,5$, the continuum loops are m…

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 $d\ge 3$, let $\theta_d(N)$ denote the one-arm probability for the loop soup $\mathcal{L}_{1/2}^{\widetilde{\mathbb{Z}}^d}$, namely … At the critical dimension $d=6$, the one-a…

Let $d>6$, and consider sign clusters of the metric graph Gaussian free field, or equivalently loop clusters of the critical loop soup. Werner's conjecture. These clusters should b…

Let $D_N\subset N^{-1}\mathbb{Z}^2\cap D$ approximate a domain $D$. Let $\mathcal{L}_{D_N}^\theta$ be a random-walk loop soup of intensity $\theta\in(0,1/2]$, let $\mathcal{C}_x$ b…

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…

Poisson–Dirichlet correlation conjecture. One has

Macroscopic-loop merge-rate conjecture. The number of endpoint pairs that allow two macroscopic loops $\gamma$ and $\gamma'$ to merge is approximately

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…

Consider the spin model on the square lattice at edge-weight $x=1/3$ and on the hexagonal lattice at edge-weight $x=1/2$. In the continuum scaling limit, consider the collection of…

Critical magnetization-field conjecture. If $x=1/3$ on the square lattice or $x=1/2$ on the hexagonal lattice, then the magnetization field, when properly rescaled, has a continuum…

In two dimensions, consider the spin field defined from the non-backtracking loop soup on the vertices of the dual graph. Spin-field scaling-limit conjecture. The scaling limit of…