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 ${\displaystyle 3\ln (d{)}^2\mathrm{/}2+9\ln (d)+9}$ and once rescaled by $\ln (d{)}^2$, is asymptotically bounded from below by $3\mathrm{/}4$. In order to get this lower bound, given disjoint isometric embeddings of a bidisc of size $1\mathrm{/}\sqrt{d}$ 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 $+\mathrm{\infty }$.