Generalized percolation games on the $2$-dimensional square lattice, and ergodicity of associated 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, independently, a label that reads trap with probability $r$ and open with probability $(1-r)$. A percolation game is played on this random board, wherein two players take turns to make moves, where a move involves relocating the token from where it is currently located, say $(x,y) \in \mathbb{Z}^{2}$, to one of $(x+1,y)$ and $(x,y+1)$. A player wins if she is able to move the token to a vertex labeled a target, or force her opponent to either move the token to a vertex labeled a trap or along an edge labeled a trap. We seek to find a regime, in terms of $p$, $q$ and $r$, in which the probability of this game resulting in a draw equals $0$. We consider special cases of this game, such as when each edge is assigned, independently, a label that reads trap with probability $r$, target with probability $s$, and open with probability $(1-r-s)$, but the vertices are left unlabeled. Various regimes of values of $r$ and $s$ are explored in which the probability of draw is guaranteed to be $0$. We show that the probability of draw in each such game equals $0$ if and only if a certain probabilistic cellular automaton (PCA) is ergodic, following which we implement the technique of weight functions to investigate the regimes in which said PCA is ergodic.

💡 Research Summary

The paper studies a family of two‑player impartial games played on the infinite two‑dimensional square lattice ℤ². Each vertex is independently labelled “trap” with probability p, “target” with probability q, or “open” with probability 1‑p‑q. Independently, each directed edge (from (x,y) to (x+1,y) or (x,y+1)) is labelled “trap” with probability r and “open” with probability 1‑r. A token starts at the origin; on her turn a player moves the token to one of the two forward neighbours, but only along edges that are open. A player wins if she either moves the token onto a target vertex or forces the opponent to move onto a trap vertex or along a trap edge. If the game never ends, it is declared a draw.

The authors ask for which triples (p,q,r) the probability of a draw is zero. They also consider two special cases: (i) a “bond” version where vertices are unlabeled and each edge can be trap, target, or open (probabilities r, s, 1‑r‑s); (ii) the original one‑parameter model obtained by setting s=0. For each model they identify regions in the parameter space where draws cannot occur.

A key conceptual step is to translate the game into a probabilistic cellular automaton (PCA). For each lattice site (x,y) one can define a state from the set {W (first player wins), L (first player loses), D (draw)} based on the states of its two forward neighbours and the local labels. The update rule is exactly the game’s recursion, and the whole lattice evolves synchronously according to this rule. The authors prove that “draw probability = 0” is equivalent to “the associated PCA is ergodic”, i.e. it possesses a unique invariant distribution and converges to it from any initial configuration.

To establish ergodicity they employ the “weight‑function” (or potential‑function) method introduced in earlier work on related PCA. A weight function V assigns a non‑negative real number to each local configuration; one shows that the expected value of V after one update step is strictly smaller than before, uniformly over all configurations. This monotone decrease forces convergence to a single invariant measure. The technique is particularly suited to regimes where traditional coupling or spectral‑gap arguments fail, such as when the parameters are extremely small.

The main results are:

1. Bond game (r,s)
   – If r = s > 0 and r ≥ 0.10883, the PCA is ergodic, so draws have probability zero.
   – If both r and s are ≤ 1/50 and satisfy (1‑s)(2s‑s²)² ≥ 4r(1‑6s+3s²), ergodicity also holds. This covers a neighbourhood of the origin where both trap and target probabilities are tiny but the target probability dominates in a precise quantitative way.

2. Three‑parameter game (p,q,r)
   Four families of inequalities are given; two particularly transparent ones are:
   – p ≤ q² and p + q ≥ 2r.
   – q + r ≤ p and 5q ≥ 4r.
   Under either set, the associated PCA is ergodic, implying zero draw probability. Consequently, even when p = q = r = ε for arbitrarily small ε > 0, draws disappear. The theorem also shows that if r = 0 (no edge traps) any positive p or q already forces ergodicity, recovering earlier results.

These findings reveal a sharp phase transition at the origin: at (p,q,r) = (0,0,0) the game is a perpetual draw with probability one, but arbitrarily close to this point any non‑zero trap or target density forces the system into an ergodic regime where a draw cannot occur. The paper thus bridges percolation theory, combinatorial game theory, and interacting particle systems.

Methodologically, the work demonstrates how carefully constructed weight functions can handle parameter regimes where the system is “almost open” and traditional percolation arguments are insufficient. It also illustrates the power of recasting combinatorial games as PCAs, turning a game‑theoretic question into a problem of stochastic dynamics.

The authors conclude by noting that their techniques may extend to higher dimensions, to games with more movement directions, or to multi‑player settings, and that the weight‑function approach could be a valuable tool for studying ergodicity in other elementary PCAs where coupling arguments are unavailable.