Some Bounds on the Bulk Conductivity of a two-phase Medium: a Comparison between Periodic and Random Structure

We prove that the effective conductivity of a three-dimensional medium with a periodic chessboard structure does not exceed\Lambda\sqrt{\alpha\beta}, where \alpha and \beta are the values of the conductivity in the cells of the chessboard, and \Lambda is a positive constant; then we show how the corresponding "random" structure behaves in a quite different way, according to recent results in percolation theory