8 problems

Let $d>8$, let $\mathbb{T}_r^d$ be the $d$-dimensional torus of volume $V$, and let $I_0(s)=\int_0^\infty e^{-t^4/4+st^2/2}\,dt$. Let $G_p^\mathbb{T}(x)$ be the torus two-point fun…

Let lattice trees (LTs) and lattice animals (LAs) be considered on a lattice, and let $d_c$ denote their upper critical dimension, namely the dimension above which their critical e…

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

Lattice-tree and lattice-animal profile conjecture. The same critical finite-size scaling profile $I_0$ occurs for both lattice trees and lattice animals on $\mathbb{T}_r^d$ above…

Let $\mathbb{G}=(\mathbb{V},\mathbb{E})$ have vertex set $\mathbb{V}=\mathbb{Z}^d$ and edge set $\mathbb{E}=\{\{x,y\}:0<\|y-x\|_\infty\le L\}$, with $d>8$ and $L$ large. A lattice…

Let $\beta_c^Q(\mathcal{P}^X)$ denote the quenched critical value for the model $\mathcal{P}^X$, with $\mathcal{T}^{\circ \! |\! \circ}$ and $\mathcal{T}^+$ the penetrable and impe…

Let Bernoulli percolation and lattice trees be defined on transitive graphs of polynomial growth, and let the volume-growth dimension denote the dimension governing the graph's pol…

Aldous's lattice-tree conjecture. If $d>8$, then $X_N$ should converge to $\mathrm{ISE}$ as $N\to\infty$.