A modified barrier method shows that the expected number of components longer than O(sqrt(d^{-1} log log d)) and nests deeper than O(log log d) in random real plane curves tends to infinity as degree d increases.
Szegö kernels and a theorem of Tian
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On Nests and Large Components of Random Real Algebraic Curves
A modified barrier method shows that the expected number of components longer than O(sqrt(d^{-1} log log d)) and nests deeper than O(log log d) in random real plane curves tends to infinity as degree d increases.