Related papers: Percolation games, probabilistic cellular automata…
Each vertex of the infinite $2$-dimensional square lattice graph is assigned, independently, a label that reads trap with probability $p$, target with probability $q$, and open with probability $(1-p-q)$, and each edge is assigned,…
Different versions of percolation games on $\mathbb{Z}^{2}$, with parameters $p$ and $q$ that indicate, respectively, the probability with which a site in $\mathbb{Z}^{2}$ is labeled a trap and the probability with which it is labeled a…
We study a one-dimensional generalized probabilistic cellular automaton $E_{p, q}$ with universe $\mathbb Z$, alphabet $\mathcal A = \{0, 1\}$, parameters $p$ and $q$ such that $0 < p+q \leq 1$ and two neighbourhoods $\mathcal N_0 = \{0,…
Consider a rooted Galton-Watson tree $T$, to each of whose edges we assign, independently, a weight that equals $+1$ with probability $p_{1}$, $0$ with probability $p_{0}$ and $-1$ with probability $p_{-1}=1-p_{1}-p_{0}$. We play a game on…
We consider the following two-player game on a graph. A token is located at a vertex, and the players take turns to move it along an edge to a vertex that has not been visited before. A player who cannot move loses. We analyze outcomes with…
We study the bond percolation game and the site percolation game on the rooted Galton-Watson tree $T_{\chi}$ with offspring distribution $\chi$. We obtain the probabilities of win, loss and draw for each player in terms of the fixed points…
We focus on a family of one-dimensional probabilistic cellular automata with memory two: the dynamics is such that the value of a given cell at time $t+1$ is drawn according to a distribution which is a function of the states of its two…
The hard-core probabilistic cellular automaton has attracted a renewed interest in the last few years, thanks to its connection with the study of a combinatorial game on percolation configurations. We provide an alternative proof for the…
In this paper, we revisit a classic example of probabilistic cellular automaton (PCA) on {0, 1} Z , namely, addition modulo 2 of the states of the left-and right-neighbouring cells, followed by either preserving the result of the addition,…
We consider a probabilistic cellular automaton (PCA) of evaporation-deposition on the one-dimensional lattice having $n$ sites with periodic boundary conditions, in which each site, during each epoch, can be in one of two states: $0$ and…
Ergodicity of probabilistic cellular automata is a very important issue in the PCA theory. In particular, the question about the ergodicity of all PCA with two-size neighbourhood, two letters alphabet and positive rates is still open. In…
We consider two-player combinatorial games in which the graph of positions is random and perhaps infinite, focusing on directed Galton-Watson trees. As the offspring distribution is varied, a game can undergo a phase transition, in which…
In this article we study a class of shift-invariant and positive rate probabilistic cellular automata (PCA) on rooted d-regular trees $\mathbb{T}^d$. In a first result we extend the results of [10] on trees, namely we prove that to every…
The positive rates conjecture states that a one-dimensional probabilistic cellular automaton (PCA) with strictly positive transition rates must be ergodic. The conjecture has been refuted by G\'acs, whose counterexample is a cellular…
We consider two-player stochastic games played on a finite graph for infinitely many rounds. Stochastic games generalize both Markov decision processes (MDP) by adding an adversary player, and two-player deterministic games by adding…
The $(m,b)$ Maker-Breaker percolation game on $(\mathbb{Z}^2)_p$, introduced by Day and Falgas-Ravry, is played in the following way. Before the game starts, each edge of $\mathbb{Z}^2$ is removed independently with probability $1-p$. After…
Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random…
A fast computer algorithm, the pebble game, has been used successfully to study rigidity percolation on 2D elastic networks, as well as on a special class of 3D networks, the bond-bending networks. Application of the pebble game approach to…
Temporal graphs are a popular modelling mechanism for dynamic complex systems that extend ordinary graphs with discrete time. Simply put, time progresses one unit per step and the availability of edges can change with time. We consider the…
A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov…