  The gas-phase metallicity distribution within galaxies records critical
information about galactic evolution. In this work we investigate how active
galactic nuclei (AGN) influence this distribution by measuring the two-point
correlation functions of gas-phase metallicity in 95 non-AGN and 37 AGN-host
galaxies from the Calar Alto Legacy Integral Field spectroscopy Area integral
field spectrographic survey. We measure metallicity using a novel Bayesian
method that properly includes both stellar and AGN contributions to emission
line fluxes and allows us to measure metallicities in both AGN-host and non-AGN
galaxies in a single, consistent framework. We find that the two-point
correlation functions of both AGN-host and non-AGN galaxies are well-fit by a
simple injection-diffusion model, and that the correlation lengths
$l_\mathrm{corr}$ we derive for the non-AGN galaxies are reasonably consistent
with those obtained in earlier work. The AGN-host galaxies generally have
smaller $l_\mathrm{corr}$ than non-AGN galaxies at fixed stellar mass, but
similar $l_\mathrm{corr}$ at fixed star formation rate (SFR), suggesting that
the primary effect of hosting an AGN in this sample is a reduction in SFR at
fixed stellar mass, and that this in turn suppresses the correlation length.
Our findings further indicate that, while both SFR and stellar mass are
positively correlated with metallicity correlation length $l_\mathrm{corr}$,
the former is more fundamental, implying that fluctuations in the metallicity
distribution within galaxies are driven more by short-term responses to
physical processes such as star formation that can change much faster than a
Hubble time.
