Probability Theory and Related Fields

, Volume 174, Issue 3–4, pp 1187–1217 | Cite as

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

  • Alexander E. Holroyd
  • Irène MarcoviciEmail author
  • James B. Martin


Let each site of the square lattice \(\mathbb {Z}^2\) be independently assigned one of three states: a trap with probability p, a target with probability q, and 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 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 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 \(d\ge 3\). 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 \(d\ge 3\), but here our method encounters a fundamental obstacle.


Combinatorial game Percolation Probabilistic cellular automaton Ergodicity Hard-core model 

Mathematics Subject Classification

05C57 60K35 37B15 



JBM was supported by EPSRC Fellowship EP/E060730/1. IM was supported by the Fondation Sciences Mathématiques de Paris. We are grateful to two referees for valuable comments. In particular, the proof of Theorem 2(ii) was prompted by a referee’s observation that such an approach gives a straightforward proof of the \(q=0\) case of Theorem 1.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Alexander E. Holroyd
    • 1
  • Irène Marcovici
    • 2
    Email author
  • James B. Martin
    • 3
  1. 1.SeattleUSA
  2. 2.Institut Elie Cartan de LorraineUniversité de LorraineVandoeuvre-lès-Nancy CedexFrance
  3. 3.Department of StatisticsUniversity of OxfordOxfordUK

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