The Size-luminosity Relation of the AGN Torus Determined from the Comparison between Optical and Mid-infrared Variability

We investigate the optical variability of low-redshift ($0.15< z\leq0.4$) active galactic nuclei using the multi-epoch data from the Zwicky Transient Facility. We find that a damped random walk model well describes the ensemble structure function in the $g$ band. Consistent with previous studies, more luminous active galactic nuclei tend to have a steeper structure function at a timescale less than the break timescale and smaller variability amplitude. By comparing the structure functions in the optical with the mid-infrared obtained from the Wide-field Infrared Survey Explorer, we derive the size of the dusty torus using a toy model for the geometry of the torus. The size of the torus positively correlates with the luminosity of the active nucleus, following a relation that agrees well with previous studies based on reverberation mapping. This result demonstrates that the structure function method can be used as a powerful and highly efficient tool to examine the size of the torus.