Estimates of critical probabilities of percolation on finite square grids

In this paper, we investigate the problem of determining the critical probabilities of percolation for finite square grids. Basing on the Harris – Kesten theorem on critical probability $p_{c} (\mathbb{Z}^{2})$ in the infinite square grid, we prove that the exact threshold of exponential decay in the infinite square grid is equal to $\frac{1}{2}$. With the help of the evaluated value of $p_{g} (\mathbb{Z}^{2})$ we show that the critical probabilities of percolation on finite square grids are arbitrarily close to $\frac{1}{2}$ when the size of a grid is large enough.