9 problems

For a finite domain $V\subset\mathbb{Z}^d$, define strong spatial mixing as follows: there exist constants $C,m\in(0,\infty)$ such that for every $\Delta\subset V$ and…

Let $q \geq 2$ be an integer, let $\beta \in [0,1]$, and consider the $q$-state antiferromagnetic Potts model on graphs of maximum degree $\Delta$. Define … Weak spatial mixing (WS…

Let $\mu$ be a distribution on configurations of a graph, and suppose it satisfies strong spatial mixing (SSM): the total variation influence of boundary pinnings on a vertex's mar…

Let $T$) be a tree of maximum degree $\Delta$, and let $q$ be the number of colors. The uniform distribution on proper $q$-colorings of $T$ is considered with strong spatial mixing…

Spatial-mixing conjecture. The unique invariant measure of every positive-rate uniformly ergodic PCA is spatially mixing.

Critical-degree conjecture. The correct value of $d^*$ is

SSM threshold conjecture. The Poisson tree almost surely exhibits SSM up to a threshold that is asymptotically

Let $\mathbf{M}$ be a branching matrix, and for a specific value $\lambda<\operatorname{WSM}(T_{\mathbf{M}})$ let $V_\lambda$ be the fixed points of the recurrences of the marginal…

Threshold conjecture for strong spatial mixing. The parameters