12 problems

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…

Equivalence conjecture. The conditions (7) and $H(a,\sigma)\leq 0$ are equivalent.

Let $k\geq 1$, let $q\in p\mathbb{N}$, and let a subgroup of finite index define periodicity on the Cayley tree. Consider periodic Gibbs measures of the homogeneous $p$-adic Potts…

Let $\zeta\in[0,1/3]$, and let $\mathcal{Z}_a$ be the set of solutions associated with the poles of the iterates of $\tau_z$ described above. Define … and let…

Non-extremality conjecture. In the majority of the gap $(\theta_1^*,\widetilde\theta_1)$, the measure $\mu_1$ is non-extreme.

Let $G=\textit{wand}$ be the fertile graph for the three-state Hard-Core model on a Cayley tree of order $k$, and let $\lambda$ be the activity parameter with critical value…

Monotonicity conjecture. For each $m=1,\dots,q-2$, the function $\theta\mapsto\alpha_m(\theta)$ is monotone increasing. This would give the monotonicity observed in the plotted exa…

Monotonicity conjecture. The function $\alpha_-(\theta)$ is monotone decreasing for all $\theta\geq\theta_{\rm c}$, while $\alpha_+(\theta)$ is decreasing for…

Let the ferromagnetic Widom–Rowlinson model with symmetric intensities be defined on a Cayley tree whose vertices have $k+1$ nearest neighbors, and consider its tree-invariant spli…

Let $k\geq 4$ and consider the ferromagnetic model with critical values $\theta'_{\rm c}(k)$, $\lambda'_{\rm cr}(k)$, $\theta_{\rm c}(k)$, and $\lambda_{\rm cr}(k)$ as defined in t…

Uniqueness conjecture. There exists a unique $p$-adic Gibbs measure corresponding to this model.

Let $\mathcal{T}^{k}$ be the Cayley tree, with $k=4$, and let $\bar{D}$ be a dimer covering of $\mathcal{T}^{k}$. Denote by $\mu^{\bar{D}+}$ and $\mu^{\bar{D}-}$ the Gibbs states c…