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\section{Measuring the clustering properties}\label{sec-methods}  There are a number of diagnostics used to quantify the local density of galactic environments, including two- and three-dimensional correlation functions, luminosity density fields, $N^{{\rm th}}$-nearest neighbors, and distance to the nearest cluster or group. We analyze blazars by measuring the spatial covariance amplitude $B$ \citep{lon79}. A significant advantage of this method is that it can be applied to projected neighbors without knowing the true three-dimensional distribution of neighboring galaxies, which would be required for density field or nearest neighbor techniques. Accurate redshifts are not available for most galaxies in the survey we use (SDSS), especially at redshifts $z>0.2$. The technique is well-established in the literature of active galaxy environments \citep[e.g.,][]{pre88,yee87,ell89,wur97} and so our results can be compared to other studies. Finally, the method has been shown to be robust independent of the magnitude limit of observations or the counting radius of neighbors. The major disadvantage of the method is that it is a statistical measurement, and so uncertainties on individial measurements are typically quite large. We mitigate this by computing $B$ for several hundred sources and analyzing the statistics of the group, rather than focusing on individual objects.   The technique for computing $B$ is briefly described here. For any population of objects as viewed on the sky in the far-field limit, its angular distribution can be approximated by:  \begin{equation}  \label{eqn-angcov}  n[\theta]d\Omega = N_g (1 + w[\theta]) d\Omega,  \end{equation}  \noindent where $n[\theta]d\Omega$ is the number of galaxies in a ring of solid angle $d\Omega$ at angular distance $\theta$ from the center of the ring. $N_g$ is the number of background galaxies in the ring, and the factor of $(1+w[\theta])$ expresses the probability of finding additional galaxies over the background level. $w[\theta]=0$ would correspond to a uniform angular distribution of galaxies in the universe (no clumping).   The standard assumption, confirmed with deep optical observations of field galaxies, is that the angular distribution of galaxies follows a power law such that:  \begin{equation}  \label{eqn-wtheta}  w[\theta] = A\theta^{1-\gamma},  \end{equation}  \noindent where $A$ is the angular covariance amplitude and $\gamma$ an index describing the slope of the power-law distribution. The amplitude of $A$ for a particular system, then, determines the degree of clustering with respect to other similarly-scaled structures in the universe.   Since this derivation has been completely general so far, we note that subscripts are used with computing the covariance amplitudes to indicate the type of object measured. For instance, $A_{gg}$ represents the galaxy-galaxy correlation amplitude, while $A_{gB}$ is the galaxy-BL~Lac correlation amplitude.   Without a distance dependence, $A$ can be computed for any point on the sky by simply integrating Equation~\ref{eqn-angcov} from 0 out to $\theta$. This gives:  \begin{equation}  \label{eqn-angint1}  \int^\theta_0 n[\theta^\prime]d\Omega = \int^\theta_0 N_g (1 + w[\theta^\prime]) d\Omega.  \end{equation}  \noindent The solid angle subtended by an angle of $2\theta$ is $\Omega=2\pi(1-cos[\theta])$; for $\theta\ll1$, this translates to a differential:  \begin{equation}  d\Omega \simeq 2\pi \theta d\theta.  \end{equation}  \noindent Integrating Equation~\ref{eqn-angint1} over the angle yields:  \begin{eqnarray}  \int^\theta_0 n[\theta^\prime] 2 \pi \theta d\theta & = & \int^\theta_0 N_g (1 + w[\theta^\prime]) 2 \pi \theta d\theta \\  2\pi \int^\theta_0 \theta^\prime n[\theta^\prime]d\theta & = & 2 \pi N_g \int^\theta_0 \theta^\prime (1 + A {\theta^\prime}^{1-\gamma}) d\theta \\  N_t \left(\frac{\theta^2}{2}\right) & = & N_g \left(\frac{\theta^2}{2} + \frac{A\theta^{3-\gamma}}{3-\gamma}\right),  \end{eqnarray}  \noindent where $N_t$ is the integrated total number of galaxies within the circle. Solving for $A$, this gives:  \begin{eqnarray}  A = \frac{N_t - N_g}{N_g} \left(\frac{3-\gamma}{2}\right) \theta^{\gamma-1}.  \end{eqnarray}  Therefore, the angular covariance amplitude can be calculated for any galaxy as a function of the total number of galaxies in the field ($N_t$), the assumed background counts from a control field ($N_g$), the power-law index $\gamma$, and the field size $\theta$. We assume a canonical value of $\gamma=1.77$. The size of the field is determined by the scales used to derive the power-law dependence of $w[\theta]$; typical values are $\theta\leq1.5^\circ$.   %\section{The spatial covariance amplitude}  For a three-dimensional distribution of galaxies around some point in space, its spatial distribution can be parameterized as:  \begin{equation}  n[r]dV = \rho_g (1 + \xi[r]) dV,  \end{equation}  \noindent where $n[r]dV$ is the number of galaxies in a spherical shell at distance $r$ from the center. $\rho_g$ is the spatial density of background galaxies in the shell, and the factor of $(1+\xi[r])$ expresses the probability of finding additional galaxies over the background level.   If the de-projected angular distribution in Equation~\ref{eqn-wtheta} is a power-law, then the spatial distribution will also follow a power-law with an index of $-\gamma$ (due to the increase in dimensions):  \begin{equation}  \xi[r] = Br^{-\gamma}.  \end{equation}  \noindent Here, $B$ is the spatial covariance amplitude, with subscripts indicating the pairs of objects for which the correlation function is computed (similar to $A$).   %\section{Converting $A$ to $B$}  \citet{lon79} project the spatial covariance function into angular space to establish a relationship between $A$ and $B$:   \begin{equation}  \label{eqn-atob}  B = \frac{A~N_{bg}[m]}{I_\gamma} \frac{D^{\gamma-3}}{\Psi[M(m,z)]}.  \end{equation}  Substituting for $A$, this gives the final form:  \begin{equation}  \label{eqn-bgb}  B = (N_t - N_{bg})\frac{(3-\gamma) D^{\gamma-3} \theta^{\gamma-1}}{2 A_\theta I_\gamma \Psi[M(m,z)]}  \end{equation}  \noindent Here, $m$ is the apparent magnitude completeness limit of the observation; depending on the redshift $z$, this is translated into an absolute magnitude limit $M(m,z)$. $D$ is the angular diameter distance\footnote{\citet{lon79} give this value as the co-moving distance. Later derivations use the luminosity distance \citep[eg,][]{yee87,ell91}; the most recent references replace it with the angular diameter distance \citep{yee99,muz07,zau07}. I have not found an explicit reference to the correction.} to the source. $\Psi[M]$ is the normalized integral luminosity function of galaxies in the field down to brightnesses of $M$. $n_{bg}[m]$ is the surface density of background galaxies brighter than $m$, and $I_\gamma$ is an integration constant dependent on the index of the power-law. For the assumed value of $\gamma$, $I_\gamma=3.87$.   Since $B$ is impossible to measure directly without explicit data on the three-dimensional positions of all objects in the field (often not possible for fields of view with many faint objects), the standard technique is to measure $A$ from deep exposures and use Equation~\ref{eqn-atob} to determine $B$. Therefore, measuring $B$ for any particular field requires:  \begin{itemize}  \item $\theta$ - angular size of the field  \item $m$ - apparent magnitude limit of the observation  \item $z$ - redshift of the target  \item $N_t$ - total number of galaxies in the field  \item $N_g$ - the expected background counts of galaxies down to $m$  \item $\Psi[m,z]$ - luminosity function of galaxies down to $m$ at redshift $z$  \end{itemize}  The uncertainty in a value of $B$ can also be calculated as a function of the total number of background and galaxy counts:  \begin{equation}  \label{eqn-deltab}  \frac{\Delta B}{B} = \frac{\sqrt{(N_t - N_{bg}) + 1.3^2 N_{bg}}}{N_t - N_{bg}} = \frac{\sqrt{N_t + 0.69 N_{bg}}}{N_t - N_{bg}}.  \end{equation}  \noindent This error estimate is considered to be conservative, including both the Poissonian error in the net counts $(N_t - N_{bg})$ and dispersion in the background counts ($N_{bg}$). The factor of $1.3^2$ is included to account for the clustered, non-Poissonian distribution of the background counts \citep{yee99}.   %\subsection{Computing $B_{gB}$}  The integrated luminosity function $\Psi[M(m,z)]$ is assumed to be in the form of a Schechter function, which is a combination of a power law at fainter luminosities and an exponential at brighter luminosities. In magnitudes, the differential Schechter LF is:  \begin{eqnarray}  \label{eqn-schechter}  \phi[M] dM = 0.4 ({\rm ln} 10) \phi^*~(10^{0.4(M^* - M)(\alpha + 1)}) \times \nonumber \\  {\rm exp}[-10^{0.4(M^*-M)}],  \end{eqnarray}  \noindent where \phistar~is the scaling for a volume-limited sample, \mstar~is the characteristic magnitude of the LF ``knee'' and $\alpha$ is the slope of the power-law portion of the LF. In theory, this function can be fit with data to determine all three parameters in a given field. In practice, \citet{yee87} and collaborators treat all three variables somewhat differently. $\alpha$ is assumed to be either $-1.0$ or $-1.2$, based on fits to previous galaxy fields; the covariance amplitude is relatively insensitive to this, since the LF is not typically not integrated down to more than two magnitudes fainter than \mstar. \mstar~is taken from previous optical surveys \citep{kin85,seb86} that computed Schechter LFs for different cosmologies and a weighted mix of galaxy morphologies. The absolute magnitudes are translated into $r$-band using color models of galaxy morphologies, and then $K$-corrected depending on the redshift of the cluster. Values for \mstar~range from $M_R\sim-22$ to $-19$ mags. Finally, the scaling value \phistar~is determined by integrating the luminosity function over the volume of the field and then scaling it to the observed control field counts at each redshift epoch.   The integrated luminosity function for computing the covariance amplitude is:  \begin{equation}  \label{eqn-schechter_int}  \Psi[M] = S_n \int_{-\infty}^{M} \phi[M^\prime] dM^\prime,   \end{equation}  \noindent where $M$ is the absolute magnitude corresponding to the observational limit of the survey. Since background galaxy counts rise much faster than the LF at magnitudes fainter than \mstar, \citet{yee87} count galaxies only up to the \underline{brighter} of two limits: the completeness magnitude of the survey ($M=m_c$) or $M=M^* + 2.5$. Since any given field is not volume-limited, the scaling factor $S_n$ normalizes the luminosity function in agreement with background galaxy counts.   If the luminosity function for each field is being computed from the data, then galaxies in the field must have their apparent magnitudes $K$-corrected to the observed waveband. For the binned redshift distribution in \citet{yee87}, a differential $K$-correction corrects all observed $r$-band galaxies in the bin to the average redshift of the group; the colors for each morphology are taken from \citet{seb86}. In addition, evolution of the LF as a function of lookback time will also affect the total number of counts in each field. The effects can be summed into the characteristic magnitude for each morphological type $i$ as a function of redshift:  \begin{equation}  \label{eqn-mstar_morph}  M_i^*[z] = M_i^*[0] + K_i[z] + E_i[z],  \end{equation}  \noindent where $M_i^*[z]$ is summed over morphology type into an integrated LF, and then integrated again over volume so that it can be directly compared to the observed galaxy counts.