Percolation games on rooted regular trees and ergodicity of associated probabilistic tree automata

We study the \emph{bond percolation game} and \emph{site percolation game} on the rooted tree $T_{d}$ in which each vertex has precisely $d$ children, for $d \geqslant 2$. In the former, each edge of $T_{d}$ is, independently, assigned a label that reads \emph{trap} with probability $p$ and \emph{safe} with probability $1-p$, for some pre-specified parameter $p \in (0,1)$, whereas in the latter, a similar random labeling is assigned to the vertices of $T_{d}$. Two players take turns to make moves, where a move involves relocating a token from where it is currently situated, say a vertex $u$ of $T_{d}$, to one of the $d$ children of $u$. A player wins the bond percolation game if she can force her opponent to move the token along an edge marked a trap, while the site percolation game is won by a player if she can force her opponent to move the token to a vertex labeled a trap. We show that in both of these games, the probability of draw is $0$ if and only if $p \geqslant p_{c}$, where $p_{c} = 1 - \frac{(d+1)^{d-1}}{d^{d}}$. We study two \emph{probabilistic tree automata}, $B_{p}$ and $N_{p}$, that represent the recurrence relations arising out of the bond percolation game and the site percolation game respectively, and we show that each of $B_{p}$ and $N_{p}$ is ergodic if and only if $p \geqslant p_{c}$.