3 problems

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…