4 problems

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…

For $q\in[1,\infty)$, let $p_c(q,h)$ be the critical edge parameter and let $\phi^1_{p,q,h,\mathbb{Z}^d}$ denote the wired random-cluster measure. The wired-percolation monotonicit…

For parameters $q\geq1$ and $h\geq0$, let $\mathfrak{Z}$ denote the relevant pressure function, and consider the phase transition as a function of $p$. The Kertész-line discontinui…

Let $p_c$ be the critical edge parameter and let $h_c(p)$ denote the Kertész line in the planar Ising case. The planar Ising Kertész-line conjecture. As $p\to p_c$, … for some cons…