## Context:

This a sequel to the question: Is the Erdős–Rényi giant component result applicable here?

Consider a matrix whose elements are independently assigned a value $1$ with probability $p$ and a value $0$ with probability $1-p$.

Define a

clusterof cells as a maximal connected component in the graph of cells with the value of $1$, where edges connect cells whose rows and columns both differ by at most $1$ (so up to eight neighbours for each cell).Is it possible to reformulate this matrix model to match the Erdős–Rényi graph model? I was wondering whether the Giant component result will be applicable for this model and whether it will be possible to reframe this problem to match the Erdős–Rényi model. I think they're related but not sure exactly how. One problem is that the $p$ here is the probability of a node being 1, rather than the probability of the existence of an edge, unlike the Erdős–Rényi model.

According to @Ben Barber's answer:

In these situations, you typically expect behaviour that, at least on a very coarse level, mirrors that observed in Erdős–Rényi random graphs. However, even for the more closely related bond (edge) percolation (where adjacent vertices are connected independently with probability $p$) the specifics of the results and the techniques used to prove them different enough that neither result is a simple application of the other.

The essential difference is that vertices in lattices start out much further apart than vertices in a complete graph $K_n$. It's easier for vertices to be isolated (as the maximum number of vertices they could possibly be adjacent to is lower) and also easier to separate off large clumps of vertices from the rest of the graph: you only need to find an empty ring of vertices surrounding a cluster, rather than an enormous bipartite subgraph which happens to contain no edges. The end result is that percolation in lattices is typically studied

at much higher valuesof $p$ than the Erdős–Rényi process—constant $p$ rather than $p \approx 1/n$—and different probabilistic tools are of use for studying the random variables that arise.

Further, @Simon L Rydin Myerson says:

On a casual inspection, I cannot actually find an example where diagonal connections (cells connected only at their corners) are permitted, as in your question.

## Question:

It seems that random matrix model, in this case, cannot exactly be modelled as an Erdos-Reyni graph.

**So, basically, my question is, can this particular random matrix model be related to any existing graph theory model leaving aside the Erdos-Reyni model (where diagonal connections are permitted)?**

**I'd actually be interested to know if there are some existing results about why giant clusters are formed in such square grids beyond a certain value of $p$ (I'm trying to make an analogy with the giant component formation in Erdos Reyni graphs).**

P.S: Please note that this is **NOT A PERCOLATION THEORY PROBLEM**. Percolation theory is concerned with the formation of spanning clusters, which is not what I meant by "giant component". A "giant component" need not necessarily be a spanning cluster. If all the cells leaving aside the edge cells get filled even that is a giant component without being a spanning cluster.

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