  We propose a new method to constrain the size of the dusty torus in
broad-line active galactic nuclei (AGNs) using optical and mid-infrared (MIR)
ensemble structure functions (SFs). Because of the geometric dilution of the
torus, the MIR response to optical continuum variations has suppressed
variability with respect to the optical that depends on the geometry (e.g.,
size, orientation, opening angle) of the torus. More extended tori have steeper
MIR SFs with respect to the optical SFs. We demonstrate the feasibility of this
SF approach using simulated AGN light curves and a geometric torus model. While
it is difficult to use SFs to constrain the orientation and opening angle due
to insensitivity of the SF on these parameters, the size of the torus can be
well determined. Applying this method to the ensemble SFs measured for 587 SDSS
quasars, we measure a torus $R-L$ relation of ${\rm log}\,R_{\rm eff}\, {(\rm
pc)} = 0.51_{-0.04}^{+0.04} \times {\rm log}\,(L_{\rm bol}/10^{46}\,\rm
erg\,s^{-1}) -0.38_{-0.01}^{+0.01}$ in the WISE $W1$ band, and sizes $\sim1.4$
times larger in the $W2$ band, which are in good agreement with dust
reverberation mapping measurements. Compared with the reverberation mapping
technique, the SF method is much less demanding in data quality and can be
applied to any optical+MIR light curves for which a lag measurement may not be
possible, as long as the variability process and torus structure are
stationary. While this SF method does not extract all information contained in
the light curves (i.e., the transfer function), it provides an intuitive
interpretation for the observed trends of AGN MIR SFs compared with optical
SFs.
