| In 1989, Burton and Keane showed under the conditions of stationarity and ergodicity, percolation in Z2 results (a.s.) in topological strips. That is, for X=x∣ x:Z2→ 0,1 , if Q is the event that x∈X contains a ribbon whose complement in Z2 has at least three components, then P(Q) = 0. Furthermore, with the added constraint of finite energy, there can be at most two infinite ribbons.;Following up those results, we construct a Z2 model in which two colors do indeed percolate under the conditions of stationarity, ergodicity and uniform finite energy (implying finite energy). This is done by constructing "yin-yang" building blocks and iteratively building up the space. The required measure is obtained by a cut and stack method.;We continue to show that the limitation of only two colors percolating does not hold in Z3 by constructing a model which has infinitely many colors percolate under the conditions of stationarity, ergodicity and uniform finite energy. Unlike the two dimensional model, we build this model from one to two to three dimensions. After extending our space to three dimensions, we induce uniform finite energy and show percolation occurs. |
