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Asymptotics in High Dimensions for Percolation

We prove that the critical probability for bond or site percolation on Z is asymptotically equal to 1/(2d) as d → ∞. If the probability of a bond (respectively site) to be occupied is γ/(2d) with γ > 1, then for the bond model the percolation probability converges as d → ∞ to the strictly positive solution y(γ) of the equation y = 1− exp(−γy). In the site model the percolation probability is asymptotically equal to γy(γ)/(2d) under these conditions. An asymptotic independence property for the random field of sites which belong to the infinite cluster is given.