14 problems

Consider broadcasting on regular grids with binary symmetric channels of crossover probability $\delta\in\big(0,\frac{1}{2}\big)$, where every vertex with two inputs uses the same…

Consider noisy one-dimensional probabilistic cellular automata, meaning one-dimensional cellular automata whose local transition probabilities are positive. The probabilistic cellu…

Let $P$) be the one-dimensional probabilistic cellular automaton on configurations with state space ${\mathbb{S}}=\{0,1\}$, neighborhood $\{0,1\}$, and transition probabilities … L…

Let $\mathcal{H}_2=\{\underline{\mathtt{0}},\underline{\mathtt{1}}\}$ be the homogeneous two-dimensional tiling space, and consider the discrete-time version of the probabilistic c…

Let $\mathcal{Y}^*=\bigcup_{k\in\mathbb{N}\setminus\{0\}}\mathcal{Y}^k$ be the set of nonempty finite strings over $\mathcal{Y}$, let $\{Y_k:k\in\mathbb{N}\}$ be the Markov chain a…

Let $P_{X_k}^{+}$ and $P_{X_k}^{-}$ denote the distributions of the observations at layer $k$ conditioned on the source bit being $1$ and $0$, respectively. Fix…

The asymptotic distribution conjecture. The probability of configuration $x$ in the steady state is

A probabilistic cellular automaton (PCA) or interacting particle system (IPS) on the one-dimensional lattice $\mathbb{Z}$ has positive rates if all its local probability transition…

Let a cellular automaton or probabilistic cellular automaton evolve configurations in the binary state space on the one-dimensional lattice $\mathbb{Z}$. The GKL cellular automaton…

Strict V-causality hierarchy conjecture. For every $k$, there exists a $V^kBox$ which is $V^k$-causal but not $V^{k+1}$-causal.

Let $\operatorname{Majority}(\alpha)$ denote the probabilistic cellular automaton Majority with parameter $\alpha$. An invariant measure is a probability measure invariant under th…

Ergodicity conjecture for one-dimensional positive-rates PCAs. All one-dimensional positive-rates PCAs are ergodic.

A probabilistic cellular automaton has state space $X=A^{\mathbb{Z}}$, where $A$ is a finite set, and updates all sites simultaneously and independently according to local transiti…

Let $\sigma\in\mathcal{S}$ be a configuration of the reversible probabilistic cellular automaton, and consider the stable configurations in which the plus spins fill a rectangular…