  We use projector Quantum Monte-Carlo methods to study the $S_{\rm tot}=1/2$
doublet ground states of two dimensional $S=1/2$ antiferromagnets on a $L
\times L$ square lattice with an odd number of sites $N_{\rm tot}=L^2$. We
compute the ground state spin texture $\Phi^z(\vec{r}) =
<S^z(\vec{r})>_{\uparrow}$ in $|G>_{\uparrow}$, the $S^z_{\rm tot}=1/2$
component of this doublet, and investigate the relationship between $n^z$, the
thermodynamic limit of the staggered component of this ground state spin
texture, and $m$, the thermodynamic limit of the magnitude of the staggered
magnetization vector of the same system in the singlet ground state that
obtains for even $N_{\rm tot}$. We find a univeral relationship between the
two, that is independent of the microscopic details of the lattice level
Hamiltonian and can be well approximated by a polynomial interpolation formula:
$n^z \approx (1/3 - \frac{a}{2} -\frac{b}{4}) m + am^2+bm^3$, with $a \approx
0.288$ and $b\approx -0.306$. We also find that the full spin texture
$\Phi^z(\vec{r})$ is itself dominated by Fourier modes near the
antiferromagnetic wavevector in a universal way. On the analytical side, we
explore this question using spin-wave theory, a simple mean field model written
in terms of the total spin of each sublattice, and a rotor model for the
dynamics of $\vec{n}$. We find that spin-wave theory reproduces this
universality of $\Phi^z(\vec{r})$ and gives $n^z = (1-\alpha -\beta/S)m +
(\alpha/S)m^2 +{\mathcal O}(S^{-2})$ with $\alpha \approx 0.013$ and $\beta
\approx 1.003$ for spin-$S$ antiferromagnets, while the sublattice-spin mean
field theory and the rotor model both give $n^z = 1/3 m$ for $S=1/2$
antiferromagnets. We argue that this latter relationship becomes asymptotically
exact in the limit of infinitely long-range {\em unfrustrated} exchange
interactions.
