Heuristics for class numbers and lambda invariants

Let $K=\mathbf Q(\sqrt{-d})$ be an imaginary quadratic field and let $\mathbb{Q}(\sqrt{3d})$ be the associated real quadratic field. Starting from the Cohen-Lenstra heuristics and Scholz's theorem, we make predictions for the behaviors of the 3-parts of the class groups of these two fields as $d$ varies. We deduce heuristic predictions for the behavior of the Iwasawa $\lambda$-invariant for the cyclotomic $\mathbf Z_3$-extension of $K$ and test them computationally.