  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.
