7 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…

Barequet–Barequet–Rote's conjecture.

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…

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 $\mathcal{L}_d$ be a $d$-dimensional lattice, and let $\epsilon^{\mathcal{L}_d}(n)$ denote the minimum perimeter among animals of size $n$ on $\mathcal{L}_d$. Minimal-perimeter…

Let $\beta_c^Q$ denote the quenched critical value for adsorption, and let the models be lattice trees and lattice animals in dimension $d$. Finite critical value conjecture. The q…