Heuristics for anti-cyclotomic $\mathbb{Z}_p$-extensions

This paper studies Iwasawa invariants in anti-cyclotomic towers. We do this by proposing two heuristics supported by computations. First we propose the Intersection Heuristics: these model `how often' the $p$-Hilbert class field of an imaginary quadratic field intersects the anti-cyclotomic tower and to what extent. Second we propose the Invariants Heuristics: these predict that the Iwasawa invariants $\lambda$ and $\mu$ usually vanish for imaginary quadratic fields where $p$ is non-split.