Probabilistic cellular automata and random fields with i.i.d. directions

Let us consider the simplest model of one-dimensional probabilistic cellular automata (PCA). The cells are indexed by the integers, the alphabet is {0, 1}, and all the cells evolve synchronously. The new content of a cell is randomly chosen, independently of the others, according to a distribution depending only on the content of the cell itself and of its right neighbor. There are necessary and sufficient conditions on the four parameters of such a PCA to have a Bernoulli product invariant measure. We study the properties of the random field given by the space-time diagram obtained when iterating the PCA starting from its Bernoulli product invariant measure. It is a non-trivial random field with very weak dependences and nice combinatorial properties. In particular, not only the horizontal lines but also the lines in any other direction consist in i.i.d. random variables. We study extensions of the results to Markovian invariant measures, and to PCA with larger alphabets and neighborhoods

Probabilistic cellular automata, invariant measures, and perfect sampling

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 chain. A classical cellular automaton is a particular case of PCA. For a 1-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to be also a PCA. Last, we focus on the PCA Majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure

Does Eulerian percolation on Z2Z^2 percolate ?

Eulerian percolation on Z 2 with parameter p is the classical Bernoulli bond percolation with parameter p conditioned on the fact that every site has an even degree. We first explain why Eulerian percolation with parameter p coincides with the contours of the Ising model for a well-chosen parameter $\beta$(p). Then we study the percolation properties of Eulerian percolation.Comment: This improves the previous version, only the status of one value for p is unknow

Can current offshore wealth management centres survive?

After recent scandals, many financial and wealth management centres are losing their allure and 'mid-shore' options are taking their place, argues Philip Marcovic

Police Officers and Personality Characteristics

In recent years, the public and media have raised their expectations for how police officers should conduct themselves but they sometimes only highlight the exceptional situations where officers made an egregious error on national news. Studying personality traits of officers may be one way to gain a more holistic perspective of the average police officer. Empathy and neuroticism were looked at in relation to job performance in 66 campus police officers. The findings did not support the hypothesis of a negative relation between neuroticism and job performance and a positive relation between job performance and empathy. However, there was a positive relation between age and impulsiveness and venturesomeness and total years as a law enforcement officer. There was also a negative relation between neuroticism and impulsiveness. Studies of officers’ immediate reactions in situations that they encounter while working should be conducted to capture the characteristics of police officers which may prove integral to the job

Percolation games, probabilistic cellular automata, and the hard-core model

Let each site of the square lattice $\mathbb{Z}^2$ be independently assigned one of three states: a \textit{trap} with probability $p$, a \textit{target} with probability $q$, and \textit{open} with probability $1-p-q$, where $0<p+q<1$. Consider the following game: a token starts at the origin, and two players take turns to move, where a move consists of moving the token from its current site $x$ to either $x+(0,1)$ or $x+(1,0)$. A player who moves the token to a trap loses the game immediately, while a player who moves the token to a target wins the game immediately. Is there positive probability that the game is \emph{drawn} with best play -- i.e.\ that neither player can force a win? This is equivalent to the question of ergodicity of a certain family of elementary one-dimensional probabilistic cellular automata (PCA). These automata have been studied in the contexts of enumeration of directed lattice animals, the golden-mean subshift, and the hard-core model, and their ergodicity has been noted as an open problem by several authors. We prove that these PCA are ergodic, and correspondingly that the game on $\mathbb{Z}^2$ has no draws. On the other hand, we prove that certain analogous games \emph{do} exhibit draws for suitable parameter values on various directed graphs in higher dimensions, including an oriented version of the even sublattice of $\mathbb{Z}^d$ in all $d\geq3$. This is proved via a dimension reduction to a hard-core lattice gas in dimension $d-1$. We show that draws occur whenever the corresponding hard-core model has multiple Gibbs distributions. We conjecture that draws occur also on the standard oriented lattice $\mathbb{Z}^d$ for $d\geq 3$, but here our method encounters a fundamental obstacle.Comment: 35 page

Wealth managers shouldn’t avoid markets with complex regulations

Complexity is the very reason why clients hire private banking services, writes Philip Marcovic