What is the Hidden Depolarization Mechanism in Low Luminosity AGN?

Millimeter wavelength polarimetry of accreting black hole systems can provide a tomographic probe of the accretion flow on a wide range of linear scales. We searched for linear polarization in two low luminosity active galactic nuclei (LLAGN), M81 and M84, using the Combined Array for Millimeter Astronomy (CARMA) and the Submillimeter Array (SMA). We find upper limits of $\sim 1 - 2\%$ averaging over the full bandwidth and with a rotation measure (RM) synthesis technique. These low polarization fractions, along with similar low values for LLAGN M87 and 3C84, suggest that LLAGN have qualitatively different polarization properties than radio-loud sources and Sgr A*. If the sources are intrinsically polarized and then depolarized by Faraday rotation then we place lower limits on the RM of a few times $10^7\, {\rm rad\, m^{-2}}$ for the full bandwidth case and $\sim 10^9\, {\rm rad\, m^{-2}}$ for the RM synthesis analysis. These limits are inconsistent with or marginally consistent with expected accretion flow properties. Alternatively, the sources may be depolarized by cold electrons within a few Schwarzschild radii from the black hole, as suggested by numerical models.