PERCOLATION OF LEVEL SETS FOR 2-DIMENSIONAL RANDOM-FIELDS WITH LATTICE SYMMETRY

Let psi(x), x is an element of R(2), be a random field, which may be v iewed as the potential of an incompressible now for which the trajecto ries follow the level lines of psi. Percolation methods are used to an alyze the sizes of the connected components of level sets {x: psi(x) = h} and sets {x: psi(x) less than or equal to h} in several classes of random fields with lattice symmetry. In typical cases there is a shar p transition at a critical value of h from exponential boundedness for such components to the existence of an unbounded component. In some e xamples, however, there is a nondegenerate interval of values of h whe re components are bounded but not exponentially so, and in other cases each level set may be a single infinite line which visits every regio n of the lattice.