  [Abridged] M31 has several times more globular clusters (GCs) than the Milky
Way. It contains a correspondingly larger number of low mass X-ray binaries
(LMXBs) associated with GCs, and can be used to investigate the GC properties
which lead to X-ray binary formation. The best tracer of the spatial structure
of M31 GCs is high-resolution imaging from the Hubble Space Telescope, and we
have used HST data to derive structural parameters for 29 LMXB-hosting M31 GCs.
These measurements are combined with structural parameters from the literature
for a total of 41 (of 50 known) LMXB GCs and a comparison sample of 65 non-LMXB
GCs. Structural parameters measured in blue bandpasses are found to show
smaller core radii and higher concentrations than those measured in red
bandpasses; this difference is enhanced in LMXB clusters and could be related
to stellar population differences. Clusters with LMXBs show higher collision
rates for their mass compared to those without LMXBs and collision rates
estimated at the core radius show larger offsets than rates estimated at the
half-light radius. These results are consistent with the dynamical formation
scenario for LMXBs. A logistic regression analysis finds that, as expected, the
probability of a GC hosting an LMXB increases with increasing collision rate
and proximity to the galaxy center. The same analysis finds that P(LMXB)
decreases with increasing GC mass at a fixed collision rate, although we
caution that this could be due to sample selection effects. Metallicity is
found to be a less important predictor of P(LMXB) than collision rate, mass, or
distance, even though LMXB GCs have a higher metallicity on average. This may
be due to the interaction of location and metallicity: a sample of M31 LMXBs
with a greater range in galactocentric distance would likely contain more
metal-poor GCs and make it possible to disentangle the two effects.

Astrophysics of Galaxies