9 problems

Non-degeneracy conjecture. For every $c\in (\kappa h_{\min},\kappa h_{\max})$, $U_n$ given $\operatorname{Conset}_n(h,c)$ lies asymptotically in $\operatorname{Gibbsnondeg}(c)$.

Fix a nonzero parameter $\lambda$, and let $Z^{\sigma_\Lambda}_{G}(\beta)$ denote the partition function of the Ising model with boundary condition $\sigma_\Lambda$. Suppose it is…

Fix parameters $\beta$ and $\gamma$, not both zero, and let $Z^{\sigma_\Lambda}_{G}(\lambda)$ denote the partition function of the 2-spin system with boundary condition…

Let $S_1$, $S_2$, and $S_3$ be positive integers or half-integers. For adjacent edges $e=\{1,2\}$ and $e'=\{2,3\}$, let $P_{12}=P_{S_1+S_2}(\vec{S}_1+\vec{S}_2)$ and…

Let a similar finite-range, ferromagnetic, translation-invariant and reflection-invariant system be obtained by replacing the Ising spins in Theorem 1 with spin variables in the Si…

Ferromagnetic comparison conjecture. The measure $\mu_{\Lambda}$ satisfies a log-Sobolev inequality with constant $\varrho$ such that

List-homomorphism product conjecture. Then

Weighted bipartite-graph product conjecture. Then

Let the background process have a unique stationary distribution. For each environment state $i\in\{1,2\}$, let $c_i(x,\eta)$ be the flip rate at site $x\in\mathbb{Z}$ for configur…