Abstract
Let ψ(x),x∈ℝ2, be a random field, which may be viewed as the potential of an incompressible flow for which the trajectories follow the level lines of ψ. Percolation methods are used to analyze the sizes of the connected components of level sets {x:ψ(x)=h} and sets {x:ψ(x)≦h} in several classes of random fields with lattice symmetry. In typical cases there is a sharp transition at a critical value ofh from exponential boundedness for such components to the existence of an unbounded component. In some examples, however, there is a nondegenerate interval of values ofh where components are bounded but not exponentially so, and in other cases each level set may be a single infinite line which visits every region of the lattice.
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Alexander, K.S., Molchanov, S.A. Percolation of level sets for two-dimensional random fields with lattice symmetry. J Stat Phys 77, 627–643 (1994). https://doi.org/10.1007/BF02179453
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DOI: https://doi.org/10.1007/BF02179453