  We use the hydrodynamical EAGLE simulation to study the magnitude and origin
of the scatter in the stellar mass - halo mass relation for central galaxies.
We separate cause and effect by correlating stellar masses in the baryonic
simulation with halo properties in a matched dark matter only (DMO) simulation.
The scatter in stellar mass increases with redshift and decreases with halo
mass. At $z = 0.1$ it declines from 0.25 dex at $M_{200, \rm DMO} \approx
10^{11}$ M$_{\odot}$ to 0.12 dex at $M_{200, \rm DMO} \approx 10^{13}$
M$_{\odot}$, but the trend is weak above $10^{12}$ M$_{\odot}$. For $M_{200,
\rm DMO} < 10^{12.5}$ M$_{\odot}$ up to 0.04 dex of the scatter is due to
scatter in the halo concentration. At fixed halo mass, a larger stellar mass
corresponds to a more concentrated halo. This is likely because higher
concentrations imply earlier formation times and hence more time for accretion
and star formation, and/or because feedback is less efficient in haloes with
higher binding energies. The maximum circular velocity, $V_{\rm max, DMO}$, and
binding energy are therefore more fundamental properties than halo mass,
meaning that they are more accurate predictors of stellar mass, and we provide
fitting formulae for their relations with stellar mass. However, concentration
alone cannot explain the total scatter in the $M_{\rm star} - M_{200, \rm DMO}$
relation, and it does not explain the scatter in $M_{\rm star} -V_{\rm max,
DMO}$. Halo spin, sphericity, triaxiality, substructure and environment are
also not responsible for the remaining scatter, which thus could be due to more
complex halo properties or non-linear/stochastic baryonic effects.
