Amoeba Measures of Random Plane Curves

We prove that the expected area of the amoeba of a complex plane curve of degree d is less than 3ln(d)^2/2+9ln(d)+9 and once rescaled by ln(d)^2, is asymptotically bounded from below by 3/4. In order to get this lower bound, given disjoint isometric embeddings of a bidisc of size 1/√(d) in the complex projective plane, we lower estimate the probability that one of them is a submanifold chart of a complex plane curve. It exponentially converges to one as the number of bidiscs grow to +∞.