Four- and five-dimensional cable-graph cluster scaling-limit conjecture

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 distribution exists and is supported on families of clusters of fractal dimension

$$1+\frac d2,$$

such that, for every sufficiently small $a$, only finitely many clusters have diameter greater than $a$. This conjecture concerns the existence and structure of the intermediate-dimensional scaling limit, where the limiting clusters contain effects from Brownian loops at multiple scales.