I have come to study Percolations on Graphs given by a known Triangulation of points:
Given a finite point set $V\in R^2$ let $G_D=(V,E)$ denote the Graph consisting of the points $V$ as vertices and $E$ a given Triangulation of $V$ as edges. For any point $Q\in V$ we call
$N_k(Q)=\{ W\in V : \text{the shortest Q-W-way in } G_D \text{ has length }k \}$
the k-Neighbourhood of Q in $G_D$.
Consider now a point $Z\in V$. Be wurthermore $0< \hat p < 1$. For all points $Y\in N_1(Z)$ we add $Y$ to the subgraph $G_s$ of $G_D$ with probability $\hat p^0=1$. Recursively we add each point $X\in N_2(Z)$ for which there exists a point $Y\in G_s$, so that $X\in N_1(Y)$, with the probability $\hat p^1<1$ to the subgraph $G_s$.
Take a look at the picture below. The white point is $Z$, the black points (and the coloured edges respectively) have been added to the subgraph. Note that all points of $N_1(Z)$ are added, because $\hat p^0=1$.
So here are my thoughts: This highly reminds me of a Galton-Watson process, which is a Markov Chain (i.e. it fullfils the Markov-Property / Memorylessness). This means the probability of a point/edge to be added to the subgraph only depends on the last state.
$\textbf{BUT}$: The probability $\hat p^k$ here depends on the total amount of steps we have already taken from the original point $Z$, which is what made me think we don't have a Markov-Chain here.
It is not really history-independent now, is it? Is there a specific name for this kind of Branching process? I am pretty new to Markov-Chains and to be honest probability is not my best field either... So thoughts or advise on how to decide what the process here is, would be really helpful!