5 problems

Let $\ell_x$ denote the loop O(1) model observable associated with a vertex $x$ on a finite graph. An even graph is a graph in which every vertex has even degree. Graham–Manolescu–…

Random-current sharp-transition conjecture. The single random current on $\mathbb{Z}^d$ has a unique sharp percolative phase transition, namely

Hexagonal-lattice random-current conjecture. All phase transitions of the random current and random-cluster model on the hexagonal lattice coincide:

Let $P_h^\emptyset$ denote the random-current measure and let $P_h^{\otimes2}$ denote the sourceless double random-current measure on $\mathbb{Z}^d$. The random-current monotonicit…

Let $G$ be an even graph. For $x\in(0,1)$, let $l_x$ be the loop-$\mathrm{O}(1)$-model on the even subgraphs of $G$, defined by … Here $\mathcal{E}_\emptyset(G)$ is the set of even…