  We prove a Weyl Law for the phase transition spectrum based on the techniques
of Liokumovich-Marques-Neves. As an application we give phase transition
adaptations of the proofs of the density and equidistribution of minimal
hypersufaces for generic metrics by Irie-Marques-Neves and Marques-Neves-Song,
respectively. We also prove the density of separating limit interfaces for
generic metrics in dimension 3, based on the recent work of
Chodosh-Mantoulidis, and for generic metrics on manifolds containing only
separating minimal hypersurfaces, e.g. $H_n(M,\mathbb{Z}_2) = 0$, for $4 \leq
n+1 \leq 7$. These provide alternative proofs of Yau's conjecture on the
existence of infinitely many minimal hypersurfaces for generic metrics on each
setting, using the Allen-Cahn approach.

Differential Geometry;Analysis of PDEs