The incipient infinite lattice tree scaling conjecture

Let $d>8$. For each $r\geq 2$, let $\vec{t}=(t_1,\ldots,t_{r-1})\in\mathbb{R}^{r-1}$ and $\vec{k}\in\mathbb{R}^{d(r-1)}$. Let $\rho^{(\mathrm{lt})}_{\vec{n}}(\vec{x})$ denote the incipient infinite lattice tree $r$-point function, using shortest-path distance as the time coordinate. The constants $A,V,v$ and $\delta\in(0,1)$ are those in (rhoscal).

Incipient infinite lattice tree scaling conjecture. There exist constants $A,V,v$ and $\delta\in(0,1)$ such that (rhoscal) holds for $\rho^{(\mathrm{lt})}$.

The conjecture asserts convergence of the incipient infinite lattice tree's moment measures to those of ICSBM above dimension $8$. The paper notes that establishing the required lattice-tree survival-probability asymptotics would be substantially involved.