8 problems

Optimal-exponent conjecture. The optimal exponent is

Imry–Ma conjecture. Adding a random magnetic field destroys the phase transition in dimension $2$, but not in dimensions $3$ and higher.

The random-field Ising chain (RFIC) is considered with disordered centered external field, and its typical configurations are compared with a configuration obtained by a renormaliz…

Let $T$ be slightly below the critical temperature $T_c$ in the three-dimensional random-field Ising model, and consider the Ising spins with plus boundary condition. Near-supercri…

Let $D$ be the dimension of the random-field Ising model, and set $D'=D-2$. The dimensional reduction conjecture. The critical exponents in dimension $D$ coincide with the correspo…

Let $\beta\to[0,infty]$ be the inverse temperature, let $Lambda earrow a Z^2$ be finite regions, and let $n$ denote the $ell^infty$ distance from a site $x$ to the boundary…

Let $\langle\sigma_0\rangle_{\lambda,h}$ denote the average infinite-volume magnetization at the origin for the two-dimensional critical random-field Ising model on $\mathbb Z^2$,…

Let $F(\lambda,h)$ be the excess free energy of the two-dimensional critical random-field Ising model, with disorder strength $\lambda\;\geqslant\;0$ and bias $h\in\mathbb R$. Let…