  Below $T_N = 1.1$K, the XY pyrochlore Er$_2$Ti$_2$O$_7$ orders into a $k=0$
non-collinear, antiferromagnetic structure referred to as the $\psi_2$ state.
The magnetic order in Er$_2$Ti$_2$O$_7$ is known to obey conventional three
dimensional (3D) percolation in the presence of magnetic dilution, and in that
sense is robust to disorder. Recently, however, two theoretical studies have
predicted that the $\psi_2$ structure should be unstable to the formation of a
related $\psi_3$ magnetic structure in the presence of magnetic vacancies. To
investigate these theories, we have carried out systematic elastic and
inelastic neutron scattering studies of three single crystals of
Er$_{2-x}$Y$_x$Ti$_2$O$_7$ with $x=0$ (pure), 0.2 (10$\%$-Y) and 0.4
(20$\%$-Y), where magnetic Er$^{3+}$ is substituted by non-magnetic Y$^{3+}$.
We find that the $\psi_2$ ground state of pure Er$_2$Ti$_2$O$_7$ is
significantly affected by magnetic dilution. The characteristic domain
selection associated with the $\psi_2$ state, and the corresponding energy gap
separating $\psi_2$ from $\psi_3$, vanish for Y$^{3+}$ substitutions between
10$\%$-Y and 20$\%$-Y, far removed from the 3D percolation threshold of
$\sim$60$\%$-Y. The resulting ground state for Er$_2$Ti$_2$O$_7$ with magnetic
dilutions from 20$\%$-Y up to the percolation threshold is naturally
interpreted as a frozen mosaic of $\psi_2$ and $\psi_3$ domains.
