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 3 ln(d)(2)/2 + 9 ln(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/ root 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 grows to +infinity.