What is the mean separation between a star and its closest neighbor in the Milky Way halo? My question is inspired by this previous question, Are there any stars that orbit perpendicular to the Milky Way's galactic plane?. In the great answer posted by @ConnorGarcia, one commenter said he would be really jealous of anybody that could see the view of the Milky Way from close to the galactic poles, since a star stream passes close to them. I think, on the flip side, they would envy us due to their near complete isolation. Space travel outside their home system would be much harder, for example. Here on the galactic plane, near the Sun, I guess the average distance to the closest nighboring star is about ~10 Ly or less. The Alpha Centauri system is about 4 Ly away, for example. Regarding the galactic halo, I guess if we picked a random star there, its closest neighbor would be probably located &gt; 100 Ly away. Is this estimate reasonable? Are there reliable figures on that?
I did some rough calculations, and 100 light-years doesn't seem to be a bad guess. If we assume that the average mass of a halo star is $\sim0.3M_{\odot}$, as would be expected for a typical IMF, and that the total stellar halo mass is $\sim10^9M_{\odot}$ (Deason et al. 2019), then we should expect there to be $\sim3.3\times10^9$ halo stars.$^{\dagger}$ The halo extends to somewhere near 100 kpc, so if we take it to be roughly spherica-ishl, this leads to an average number density of
$$n\approx\frac{3.3\times10^9\;\text{stars}}{\frac{4\pi}{3}(100\;\text{kpc})^3}\approx7.9\times10^{-7}\;\text{pc}^{-3}$$
and a mean inter-star distance of $l\approx n^{-1/3}\approx108\;\text{pc}$. So my back-of-the-envelope result differs from yours by only a factor of 3.
Now, the halo has a distinctly non-uniform density. It follows a power law of $r^{-n}$, with an index somewhere near $n\approx3$. Different surveys have yielded values ranging from $2.5\lesssim n\lesssim 3.5$ (ish), depending on the methodology and groups studied. The upshot is that in the outer halo, you'll certainly see much larger separations, whereas closer to the Galactic plane, you could see densities perhaps within a couple orders of magnitude of disk populations.

$^{\dagger}$Two things: First, the halo population's mass function should be different than a typical IMF, because it's composed largely of aging Population II stars - but I think $0.3M_{\odot}$ is still reasonably close. Second, I also checked the number by actually integrating via a Kroupa IMF (normalizing by total halo mass), (and again, maybe not a totally accurate distribution) and found a value within a factor of 2 of 3.3 billion.