6 problems

Consider a general ferromagnetic SOS model with coupling $J<0$, temperature $T>0$, and an integer height parameter $m>1$. A translation-invariant splitting Gibbs measure is a Gibbs…

Let $m,k\geq 2$, $J>0$, and $\beta\geq 0$. A translation-invariant splitting Gibbs measure is a Gibbs measure invariant under the tree translations. Antiferromagnetic uniqueness co…

Let $m,k\geq 2$ and $J<0$. Let $h=(h_0,h_1,\ldots,h_{m-1})$ be a vector of boundary-law parameters satisfying equation (2.6), with $h_0=0$. Symmetric-solution conjecture. For all…

Let $k\geq 2$, $J<0$, and $\beta>\beta^1_{\rm cr}$. Let $\mu^*_+$ be the top symmetric translation-invariant splitting Gibbs measure, and let $\omega^{(n)}$ denote the boundary con…

Let $m=2$, $J<0$, and $\beta\leq\beta^1_{\rm cr}$. Let $\mu^*$ denote the symmetric Gibbs measure associated with the unique solution in the ferromagnetic SOS model. Uniqueness con…

Let $m=2$, $k\geq 2$, and $J<0$. Let $\mu^*_{-}$, $\mu^*_{\rm m}$, and $\mu^*_{+}$ be the three symmetric TISGMs, and let $\delta_\omega$ denote the Dirac measure at a configuratio…