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While I was discussing this question with @JamesMartin, he mentioned a result here that: In a $n\times n$ finite square grid, if $p\geq p_c+\epsilon$, such that $\epsilon>0$ and $p_c$ is the ...

Van Den Berg-Kesten-Reimer inequality Let $n$ be a positive integer. For $i\in[n]$, let $\Omega_i$ be a finite set and $\mu_i$ a probability measure on it. Set $\Omega=\Omega_1\!\times\!\ldots\!\...

Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...

Suppose I have a $n \times n$ square grid and for each square, I assign 1 with probability $\frac{1}{2}$ and 0 with probability $\frac{1}{2}$. On the boundary, I put 1s on the lower half and 0s on the ...

This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix? In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of ...