On Nests and Large Components of Random Real Algebraic Curves

We develop a variant of the barrier method in order to address questions about topology of Kostlan random real algebraic plane curves. In particular we prove that the expected number of connected components of the curve of length at least ${\displaystyle O\left(\sqrt{d^{-1}\log \log d}\right)}$ grows to infinity with $d$, and likewise, the expected number of nests of the curve of depth at least ${\displaystyle O(\log \log d)}$ grows to infinity with $d$. In another direction, we adapt an $L^{\mathrm{\infty }}$-norm bound result of Shifmann and Zelditch to subspaces and employ it to obtain a lower bound for the probability that a finite number of points remain all in different components of the complement of a large degree random curve.