\section{INTRODUCTION}

Galaxy merging is both predicted and observed to play an important role in galaxy growth and evolution,
a number of groups studying the high-redshift universe have
found that the early type galaxy population is growing in stellar mass
over time \citep{bell04a, willmer05a, blanton06a, brown07a,
faber07a, Zhu11a}, especially at
very high ($z \sim 2$) redshift \citep{Daddi05a, Trujillo07a, Longhetti07a, Toft07a, conselice07a, Cimatti08a, vanDokkum08a, Saracco09a},
although it appears to be
growing mainly at luminosities around $L^{*}$ and below \citep{bundy06a, cimatti06a, wake06a, brown07a,
scarlata07a, cool08a}.
These red galaxies show low levels of star formation, so star formation is unlikely to provide the growth.
The number density of massive blue galaxies is far lower than that of red galaxies, so transformations
of massive blue galaxies is unlikely to provide the growth either.

In the CDM paradigm for structure formation,
galaxies reside in mass concentrations that are built from
the merging and accretion of smaller concentrations over cosmic
time.
Some recent merger rates have been predicted in theories of galaxy
formation in a cosmological context \citep{Murali02a, maller06a, conroy07a, Stewart09a}.

There are many galaxy--galaxy merger rate estimates in the
literature. Some involve identifying a class of premerger close
pairs \citep{carlberg94a, carlberg00a, patton97a, patton00a,
van Dokkum99a, lin04a, bell06a, kartaltepe07a, Masjedi06a, Masjedi08a,
Patton08a, kitzbichler08a, Bundy09a, De Propris07a, De Propris10a, Robaina10a}.
One makes use of a class of postmerger galaxies based on
star formation indicators \citep{quintero04a}.
Another relies on finding
merging sources based on disturbed or merging morphologies
\citep{abraham96a, conselice03a, van Dokkum05a, lotz06a, De Propris07a, Lotz08a, Conselice09a}.
In each case, the
estimate of the merger rate proceeds by estimating the abundance
of the class and some time interval over which they remain identifiably
part of that class.
We have found previously that luminous red galaxies (LRGs) are growing on average by
less than 2 $h_{70} \, \unit{percent \, per \, Gyr}$, from merger activity at
redshifts $0.16 < z < 0.30$ \citep{Masjedi08a}.

Galaxies are found to separate into two distinct regions in color-magnitude space: the red sequence
and the blue cloud \citep{Strateva01a, Blanton03b}. In this paper,
we consider galaxies on both sides of this divide.
We estimate the merger rate with close pairs by cross-correlating two different galaxy sets:
galaxies from NYU VAGC Spectroscopic Sample and galaxies from SDSS Imaging Sample.
Our technique for measuring the close pairs builds on our previous work \citep{Masjedi06a, Masjedi08a}.
With this technique, we can measure projected correlation functions on extremely small
scales (kpc to Mpc scales) without the need for complete spectroscopic samples, and we can also
go to much fainter sources in the imaging sample than we could if we required spectra for everything.
Furthermore, our results will not be affected by projection effects, because cross-correlation removes by-chance interlopers naturally.
We go beyond our previous work by considering a range of galaxy
masses and colors that is far broader than just the LRG Sample.
In this paper, we measure the correlation functions and growth rates for galaxies of the full range of properties
to which the SDSS main Sample \citep{Strauss02a} is sensitive.

The primary uncertainty in turning a deprojected three-dimensional cross-correlation function
at small scales into a merger rate is in estimating the mean time for two galaxies to merge as a
function of galaxy masses and separation. There are different estimates of merger timescale:
free-fall time, orbital time, and dynamical friction time
(\latin{e.g.}, \citealt{Binney87a, Boylan-Kolchin07a, conroy07a, kitzbichler08a, Bundy09a, Loz10a}).
In this paper, we will use an approximation to the dynamical friction time as our
standard estimate. In addition, we will use an orbital time for comparison.
Even the (longer) dynamical friction time is likely to be an under-estimate of the mean
merger time, because some close pairs will not merge at all. Any under-estimate of the merger
time leads to an over-estimate of the growth rate.

Throughout this paper, all magnitudes are AB, all masses are stellar masses $h_{70}^{-2}\,\mathrm {M_{\odot}}$,
all velocities are $\mathrm {km} \, \mathrm {s}^{-1}$,
all error bars showed in our figures are from the the jackknife error covariance matrix only,
and all volumes and distances are comoving, calculated for a cosmological world
model with $(\Omega_m,\Omega_{\lambda}) = (0.3, 0.7)$ and Hubble constant
$H_0 \equiv 70 \, h_{70}\, \mathrm {km} \, \mathrm {s}^{-1} \, \mathrm {Mpc}^{-1}$.

\section{DATA}

We use the NYU Value-Added Galaxy Catalog (VAGC) V7.2 data \citep{blanton05a},
which is built from SDSS data, as our source of spectroscopic data. We use all the SDSS imaging data
as our source for cross-correlation samples.
The SDSS is an imaging and spectroscopic survey of about $10^4$ square degrees
\citep{fukugita96a, gunn98a, gunn05a, York99a, York01a, stoughton02a, abazajian03a, abazajian04a}.

   \subsection{Spectroscopic Subsample}

\label{Spectroscopic Subsample}

We use a spectroscopic sample drawn from NYU VAGC V7.2 data. It contains about $8.6 \times 10^5$ main galaxies,
of which about $5 \times 10^5$ SDSS Main Samples galaxies \citep{Strauss02a} are in the redshift range $0.03 < z < 0.15$ (see Figure~\ref{fig:decVSra}).
We use the code {\tt sdss\_kcorrect} \citep{Blanton07a} to calculate
the $K$-corrected absolute magnitude at $z = 0.1$ and stellar mass for each.
We cut this sample into 10 subsamples by stellar mass and make sure that in each subsamples
the number of galaxies is the same.
After that, we cut them into red and blue using the cut \citep{hogg04a}:
\begin{equation} \label{eq:colorcut}
^{0.1}g - ^{0.1}r = 0.0625 \, \times \, \log \, (M) \, + \, 0.15
\end{equation}
where $M$ is the stellar mass.
These cut the spectroscopic sample into 20 spectroscopic subsamples; we call these subsamples $\set{D}_s$
(see Figure~\ref{fig:colorVSmass}).
For the red spectroscopic subsamples, we name the smallest stellar-mass subsample `red0', then name the second smallest stellar-mass subsample `red1',
and so on, so the largest stellar-mass subsample is named `red9', and the same for the blue spectroscopic subsamples.
Figures~\ref{fig:densityred} and \ref{fig:densityblue} show the number density of galaxies of the 18 spectroscopic subsamples
as a function of redshift.
The red0 and blue0 subsamples are low in number density, and highly affected by survey selection effects,
so we do not use them further.
We can see that blue8, blue9 and red9 appear to rise in number density with redshift,
this is because we removed galaxies with $r > 14 \, \mag$.

In the SDSS, spectroscopic targets were assigned to spectroscopic fiber plug plates with
a tiling algorithm that ensures nearly complete samples \citep{blanton03a}. The angular
completeness is characterized for each unique region of overlapping spectroscopic plates
(sector) on the sky. An operational constraint of the SDSS spectrographs, however, is that
the physical size of the fiber coupling restricts the angular separations of targets on any individual
spectroscopic plate to be larger than 55 arcsec. This fiber collision constraint causes about
9 percent of target galaxies not to have measured redshifts. Because this
project involves cross-correlating spectroscopic and imaging objects, this fiber collision limit
only comes into our analysis in our weighting scheme to account for incompleteness;
it does not affect our pair counts directly, unless the unmeasured galaxies have very different cross-correlation
amplitude with fainter galaxies than the measured galaxies (this is not likely).

For each galaxy $j$ in the spectroscopic subsample we compute a weight $p_j$ that accounts
statistically for the spectroscopic incompleteness coming from fiber collisions. We calculate
this weight by running a two-dimensional friends-of-friends grouping algorithm on the SDSS
target parent sample in the NYU VAGC V7.2, with a 55-arcsec linking length. This procedure emulates
the SDSS tiling algorithm \citep{blanton03a}. Within each collision group made by
the friends-of-friends algorithm, we find the number of objects with measured spectroscopic
redshifts and divide by the total number. The inverse of this ratio is a weight $p_j$ assigned to
each spectroscopic galaxies to account for survey incompleteness. All $p_j \ge 1$.

We have created large catalogs of randomly distributed points for each of the spectroscopic
subsamples based on the SDSS subsample angular and radial (redshift distribution) models.
These catalogs match the redshift distribution of the corresponding spectroscopic subsample
and are isotropic within the survey region.
In order to exactly match the redshift distribution, for each galaxy in each spectroscopic
subsample, we created 10 galaxies with the exactly same redshift but angular position taken from
the random sample.
Thus each spectroscopic subsample $\set{D}_s$ obtains a random catalog which is 10 times as large as $\set{D}_s$.
These catalogs provide a homogeneous baseline (expected numbers)
for the tests that follow. We call them $\set{R}_s$.

For each random point $j$ we compute a weight $f_j$ that accounts for the incompleteness
of the spectroscopic survey in that point's region of the sky \emph{not} due to fiber collision but
due to all the other selection effects in the survey. The NYU VAGC V7.2 provides the
angular geometry of the spectroscopic survey expressed in terms of spherical polygons. The
geometry is complicated: the spectroscopic plates are circular and overlap, while the imaging
is in long strips on the sky, and there are overlap regions for some plates that have not yet
been observed. The resulting spherical polygons track all these effects \citep{blanton05a} and characterize the
geometry in terms of sectors, each being a unique region of overlapping spectroscopic
plates. In each sector, we count the number of possible targets (LRG, Main, and Quasar),
excluding those missed because of fiber collisions, and the number of these whose redshifts
were determined. We weight the points in the random catalog matched to the spectroscopic
galaxies by the ratio of these numbers $f_j$. All $f_j \le 1$.

For each spectroscopic sample `s' ($\set{D}_s$ or $\set{R}_s$), in which
there are $N_s$ spectroscopic galaxies,
we divide it into the 50 bins by lines of constant dec, so that
there are $N_s/50$ spectroscopic galaxies in each bin.
Then we resample them into 50 leave-one-out resampling samples, so that
there are $49/50 \times N_s$ spectroscopic galaxies in each sample.
We call them the `50 resampling samples',
with which we can calculate our jackknife
resampling covariance matrix.

   \subsection{Imaging Subsamples}

For our imaging data, we use a sample drawn from the full SDSS imaging catalog in which there are about
$1 \times 10^8$ galaxies. We include from the SDSS imaging sample only galaxies with
apparent magnitude $14 < r < 21.5 \, \mag$, and
apparent color $ -0.5 < [g-r] < 2 \, \mag$, see Figures~\ref{fig:rcut} and \ref{fig:colorcut}.
As with the spectroscopic sample, we have created large catalogs of randomly distributed points, with the same angular distribution
as the imaging data subsamples, we call them $\set{R}_i$.

   \subsection{Grid Method $K$-corrections for galaxies from the imaging sample}

We cannot $K$-correct individual galaxies in the imaging subsamples once and for all,
because we do not have spectroscopic redshifts for them. Each time we consider a pair
of galaxies, one from the spectroscopic subsample and one from the imaging subsample, we
assign the spectroscopic redshift to the galaxy from the imaging sample. This allows us to calculate
for each galaxy from the imaging sample in each spectroscopic--imaging pair a temporary $K$-corrected stellar mass
and [$^{0.1}g \, - \, ^{0.1}r$] color for the purposes of that pair. We discard
these values and compute new ones when the galaxy from the imaging sample is used in another pair with
another galaxy from the spectroscopic sample.

To save time, we take galaxies from the NYU VAGC spectroscopic sample as representative
of all galaxy types, and apply the code {\tt sdss\_kcorrect} on the galaxies from a grid named `B'
of observed $r$-band magnitude
(0.5 mag per bin), [$g - r$] color (0.1 mag per bin), and redshifts between 0.03 and
0.15 (0.0002 per bin). We saved the mean $K$-corrected stellar mass $M_B$ and [$^{0.1}g_B \, - \, ^{0.1}r_B$] color
in a grid of observed $r$-band magnitude, [$g - r$] color, and redshift, also save the mean redshift $z_B$, mean $r_B$ and mean [$g_B - r_B$] color,
Thereafter we estimated
the $K$-corrected stellar mass and [$^{0.1}g \, - \, ^{0.1}r$] color for a
galaxy G in grid B, and the [$g_G - r_G$] color of galaxy G is between [$g_B - r_B$] color of grid B and [$g_C - r_C$] color of grid
C which is next to grid B (that means $z_B$ and $z_C$ are the same, and $r_B$ and $r_C$ are the same):
\begin{equation} \label{eq:mass}
\log \, {M_G} = \log \, {(M_B \, \times \, \frac{(d_L^G)^2}{(d_L^B)^2}) } \, - \, \frac{r_G - r_B}{2.5} \, + \, \frac{ (\log \, {M_C} \, - \, \log \,{M_B}) \times ([g_G - r_G] - [g_B - r_B])} {[g_C - r_C] - [g_B - r_B]}
\end{equation}
\begin{equation} \label{eq:color}
[^{0.1}g_G \, - \, ^{0.1}r_G] = [^{0.1}g_B \, - \, ^{0.1}r_B] + [g_G - r_G] - [g_B - r_B]
\end{equation}
where $d_L$ is the luminous distance calculated from redshift, and galaxy G is in grid B,
so the difference between $z_B$ and $z_G$ is small, similarly, the difference is small between
$d_L^G$ and $d_L^B$, $r_b$ and $r_G$, [$g_B - r_B$] and [$g_G - r_G$].
because of these small differences, the equations (\ref{eq:mass}) and (\ref{eq:color}) can be used.
This speeds up the $K$-correction procedure immensely and only introduces a 12-percent one-sigma error for each
galaxy and there is little bias (see Figure~\ref{fig:stellarmass}),
so only introduces percent-level errors in the results.
We call this `Grid Method' hereafter.

For grids with observed $r$-band magnitude $r > 17.77 \, \mag$, we cannot get the mean stellar mass and
[$^{0.1}g \, - \, ^{0.1}r$] color directly, because there are
no galaxies from the spectroscopic sample at observed $r$-band magnitude $r > 17.77 \, \mag$.
In order to estimate the color and stellar mass of the galaxies in grid A with mean observed $r$-band magnitude
$r_A > 17.77 \, \mag$, we find the grid point B that has the nearest mean observed $r$-band magnitude
$r_B$ ($r_B > 17.77 \, \mag$) and has the same observed
[$g - r$] color and the same redshift, also there are at least 10 galaxies from the spectroscopic sample in grid B. Then we estimate
$M_G$ and $[^{0.1}g_G \, - \, ^{0.1}r_G]$ color of galaxy G in grid A using the equation (\ref{eq:mass}) and (\ref{eq:color}).
If we cannot find a grid B satisfying the conditions, we will leave all the galaxies in grid A empty.

Figure~\ref{fig:color} shows the difference between the color estimate using the Grid Method above
and the color calculated using the code {\tt sdss\_kcorrect}. The one-sigma error for each
galaxy is about $0.04 \, \unit{mag}$ and there is little bias (see Figure~\ref{fig:color}).

   \subsection{Velocities for merger rate estimates}

The average orbital velocity for a galaxy from the imaging sample around a more massive
red galaxy from the spectroscopic subsample $s$ with average velocity dispersion $\sigma_v$ is very roughly $1.5$ times the
velocity dispersion. We estimate the $\sigma_{v}$ with
Faber-Jackson relation:
\begin{equation} \label{eq:velocity_red}
\log \, \sigma_{v} \, = \, a_1 \, + \, b_1 \, \log \, M_s^{red}
\end{equation}
where $M_s^{red}$ is the mean stellar mass of the red galaxies from
spectroscopic subsample $s$. We performed a linear fit to the data
to obtain $a_1 = -1.588$ and $b_1 = 0.354$ (Figure~\ref{fig:velocity}).

We estimate the average orbital velocity $V_{c}$ for a galaxy from the imaging sample around a more massive
blue galaxy from the spectroscopic subsample $s$ with the
Tully-Fisher relation:
\begin{equation} \label{eq:velocity_blue}
\log \, V_{c} \, = \, a_2 \, + \, b_2 \, \log \, \left<L_I\right>
\end{equation}
where $\left<L_I\right>$ is the mean I-band luminosity of the blue galaxies from
spectroscopic subsample $s$, caculated from $L_I = M/r_I$, where $M$ is the stellar mass of the galaxy and
$r_I$ is the I-band mass-to-light ratio of the galaxy calculated from the code {\tt sdss\_kcorrect}.
We used for this relationship $a_2 = -0.835$ and $b_2 = 0.291$ \citep{courteau07a}.

Table \ref{tab:data0} provides this information for all 20 spectroscopic subsamples.

\section{Method}

As we describe in this Section, we cross-correlate galaxies in each spectroscopic subsample $s$
with galaxies in each imaging subsample $i$ (we will define below the imaging subsamples, which are related to
each spectroscopic subsample $s$) to obtain the real-space, projected
cross-correlation function $w_{si}(r_p)$ as a function of tangential projected separation $r_p$.
We de-project the smooth fit for each cross-correlation function to obtain the true, three-dimensional,
real-space cross-correlation function $\xi_{si}(r)$ as a function of real-space separation $r$. We use
these three-dimensional cross-correlation functions and time scales from galaxy dynamics to place limits on the
accretion rate of objects from
subsample $i$ onto objects from subsample $s$, and therefore obtain the fractional mass growth rates of
the different types of galaxies.

For each spectroscopic subsample $s$, we cut the imaging sample into 16 subsamples by stellar mass:
$10^{-\frac{j}{8}} < M_i^{G}/M_s < 10^{-\frac{j-1}{8}}$, where $M_i^{G}$ is the stellar mass
of a galaxy from the imaging sample, calculated by the Grid Method using the redshift of the
spectroscopic galaxy, and $M_s$ is the mean stellar mass of the galaxies from
spectroscopic subsample $s$, and
$j$ is an integer $1 \le j \le 16$.
After that, we cut them into red and blue using equation \ref{eq:colorcut} by
$[^{0.1}g \, - \, ^{0.1}r]$ color calculated by Grid Method.
For each spectroscopic subsample $s$, these cut the imaging sample into 32 imaging subsamples.

   \subsection{Projected cross-correlation function}

The three-dimensional real-space
cross-correlation function $\xi_{si}(r)$ of two subsamples of galaxies $s$ and $i$, is defined to be the
excess probability of finding a galaxy from subsample $s$ at a three-dimensional distance $r$ from a galaxy from
the subsample $i$, relative to the ``null'' Poisson prediction. If we define $\mathrm{d} N_{si}$ to be the
infinitesimal number of galaxies expected to be found from
subsamples $s$ and $i$ separated by a distance $r$, within the
infinitesimal comoving volumes $\mathrm{d} V_s$ and $\mathrm{d} V_i$, then:
\begin{equation} \label{eq:Nsi}
\mathrm{d} N_{si} = n_s \, n_i \, [1 \, + \, \xi_{si}(r)] \, \mathrm{d} V_s \, \mathrm{d} V_i
\end{equation}
where $n_s$ and $n_i$ are the three-dimensional comoving number
densities of subsamples $s$ and $i$ respectively.

The projected two-dimensional cross-correlation function $w_{si}(r_p)$ is related to
$\xi_{si}(r)$ by an integral along the line of sight (\latin{e.g.}, \citealt{davis83a}):
\begin{equation} \label{eq:wp}
w_{si}(r_p) = 2\int_0^{\infty} \mathrm{d} {y} \, \xi_{si} \, [ (r_{p}^2 \, + \, y^2) ^{1/2} ]
\end{equation}
Because the correlation function $\xi_{si}$ is very large at small scales, in practice
this integral is dominated by scales $y \lesssim r_p$.

Following a similar approach to the one we have used previously \citep{Masjedi06a, Masjedi08a}, we estimate not $w_{si}(r_p)$ but
$\rho_i \, w_{si}(r_p)$, where $\rho_i$ is the average comoving three-dimensional stellar mass density of the imaging
subsample $i$, we estimate this by:
\begin{equation} \label{eq:mwp}
\rho_{i} \, w_{si}(r_p) = \frac{D_{s}D_{i}}{D_{s}R_{i}} - \frac{R_{s}D_{i}}{R_{s}R_{i}}
\end{equation}
where $\set{D}_s$ and $\set{D}_i$ denote the spectroscopic and imaging data subsamples, and $\set{R}_s$ and $\set{R}_i$
denote the random catalogs matched to the spectroscopic and imaging subsamples.
In a rough sense, the first term $D_{s}D_{i}/D_{s}R_{i}$ measures the mass-weighted abundance of pairs,
and the second term $R_{s}D_{i}/R_{s}R_{i}$ subtracts the mean background level.
Each of these components is explained in detail below.
By making random spectroscopic samples with the same redshift
distribution as Main Galaxy spectroscopic subsamples and cross-correlating this sample with the imaging
subsamples, we subtract the scaled random interlopers --- galaxies not at the same redshift but
treated as such by the algorithm, from our data-data correlation statistically.

The procedure described here has been tested with simulations and shown to deliver
an unbiased measure of the correlation function \citep{Masjedi06a}.
We measure the product $\rho_i \, w_{si}(r_p)$, for the purposes of estimating the
mass accretion rate, we need only this product, and not either quantity separately.

In detail, the factors are defined as follows:
\begin{equation} \label{eq:dd}
D_{s}D_{i} = \frac{\displaystyle\sum_{j \in \set{D}_s\set{D}_i}p_j}%
              {\displaystyle\sum_{j \in \set{D}_s}p_j} \quad
\end{equation}
where the top sum is over pairs $j$ with one member taken from the spectroscopic subsample
and one from the imaging subsample in some bin of transverse radii $r_p$, the bottom sum is over
galaxies $j$ from the spectroscopic subsample, and $p_j$ is the weight given to the
galaxies from the spectroscopic sample that accounts for fiber-collision incompleteness as described above. Being a
sum of dimensionless weights, this factor $D_{s}D_{i}$ is dimensionless.
\begin{equation} \label{eq:drm}
D_{s}R_{i} = \frac{\displaystyle\sum_{j \in \set{D}_s\set{R}_i}p_j}%
              {\displaystyle\sum_{j \in \set{D}_s}p_j
\,\left[\frac{\mathrm{d} \Omega}{\mathrm{d} A}\right]_j\,\frac{\mathrm{d} M}{\mathrm{d} \Omega}} \quad
\end{equation}
where the top sum is over pairs $j$ with one member taken from the spectroscopic subsample
and one taken from the random catalog matched to the imaging subsample, the bottom sum
is over galaxies $j$ in the spectroscopic subsample, $\biggl(\frac{\mathrm{d} \Omega}{\mathrm{d} A} \biggr)_j$
is the inverse square of the transverse comoving distance \citep{hogg99a} to galaxy $j$ from the spectroscopic sample, and
$\frac{\mathrm{d} M}{\mathrm{d} \Omega}$ is calculated by $\frac{\mathrm{d} M}{\mathrm{d} \Omega} = \frac{\mathrm{d} N}{\mathrm{d} \Omega} \times M_i$,
where $\frac{\mathrm{d} N}{\mathrm{d} \Omega}$ is the two-dimensional number
density of the random imaging catalog per solid angle,
\begin{equation} \label{eq:mims}
M_i = M_s \times 10^{-\frac{j \, - \, 0.5}{8}}
\end{equation}
where $M_i$ is the mean stellar mass
of galaxies from the imaging subsample $i$, and $j$ is an integer $1 \le j \le 16$.
The product of these two derivatives gives the average mass of
random imaging objects per unit comoving area around each galaxy from the spectroscopic sample, so
this factor $D_{s}R_{i}$ has dimensions of comoving area divided by stellar mass.
\begin{equation} \label{eq:rd}
R_{s}D_{i} = \frac{\displaystyle\sum_{j \in \set{R}_s\set{D}_i}f_j}%
              {\displaystyle\sum_{j \in \set{R}_s}f_j} \quad
\end{equation}
this is similar to equation (\ref{eq:dd}), but $\set{R}_s$ represents now the random catalog mentioned in Section
\ref{Spectroscopic Subsample},
and $f_j$ is the weight given to the random point in pair $j$ that accounts for the incompleteness of the
spectroscopic survey in that point's region of the sky not due to fiber collisions, as described
above. This factor $R_sD_i$ is dimensionless.
\begin{equation} \label{eq:rrm}
R_{s}R_{i} = \frac{\displaystyle\sum_{j \in \set{R}_s\set{R}_i}f_j}%
              {\displaystyle\sum_{j \in \set{R}_s}f_j
\,\left[\frac{\mathrm{d} \Omega}{\mathrm{d} A}\right]_j \frac{\mathrm{d} M}{\mathrm{d} \Omega}} \quad
\end{equation}
this is similar to equation (\ref{eq:drm}), but $\set{R}_s$ represents now the random catalog mentioned in Section
\ref{Spectroscopic Subsample},
and $f_j$ is the same to that of equation (\ref{eq:rd}).
This factor $R_{s}R_{i}$ has dimensions of comoving area divided by stellar mass.

For some experiments, we need to estimate $n_{i} \, w_{si}(r_p)$, where $n_{i}$ is the average comoving three-dimensional number density of the imaging
subsample $i$. We estimate this by the following estimation:
\begin{equation} \label{eq:nwp}
n_{i} \, w_{si}(r_p) = \frac{D_{s}D_{i}}{[D_{s}R_{i}]_N} - \frac{R_{s}D_{i}}{[R_{s}R_{i}]_N}
\end{equation}

For $D_sD_i$ and $R_sD_i$, it is as the same as equation (\ref{eq:dd}) and equation (\ref{eq:rd}). For $[D_sR_i]_N$ and $[R_sR_i]_N$:
\begin{equation} \label{eq:dr}
[D_{s}R_{i}]_N = \frac{\displaystyle\sum_{j \in \set{D}_s\set{R}_i}p_j}%
              {\displaystyle\sum_{j \in \set{D}_s}p_j
\,\left[\frac{\mathrm{d} \Omega}{\mathrm{d} A}\right]_j\,\frac{\mathrm{d} N}{\mathrm{d} \Omega}} \quad
\end{equation}
this is similar to equation (\ref{eq:drm}), but $\frac{\mathrm{d} N}{\mathrm{d} \Omega}$ is the two-dimensional number
density of the random imaging catalog per solid angle.
This factor $D_{s}R_{i}$ has dimensions of comoving area.
\begin{equation} \label{eq:rr}
[R_{s}R_{i}]_N = \frac{\displaystyle\sum_{j \in \set{R}_s\set{R}_i}f_j}%
              {\displaystyle\sum_{j \in \set{R}_s}f_j
\,\left[\frac{\mathrm{d} \Omega}{\mathrm{d} A}\right]_j \frac{\mathrm{d} N}{\mathrm{d} \Omega}} \quad
\end{equation}
this is similar to equation (\ref{eq:rrm}).
This factor $R_sR_i$ has dimensions of comoving area.

To compute the factors, we bin all the spectroscopic-imaging pairs according
to the color and the stellar mass of the galaxies from the imaging sample which are estimated by the Grid Method,
and the comoving projected separation $r_p$ of the pair
where $r_p = r_k = 0.0143 \times 10^{k/5} \, h_{70}^{-1}\,\mathrm {Mpc}$ and $k$ is an integer $0 \le k \le 14$.
As described above, the Grid Method and separations are computed
for each pair using the redshift of the galaxy from the spectroscopic sample.
We have combined the 16 stellar mass bins into 4 to simplify the figures.
Figures~\ref{fig:nwpredWITHred} through \ref{fig:nwpblueWITHblue} show the
results of our measurements of $\rho_{i} \, w_{si}(r_p)$.

The uncertainties on the results shown in these figures are estimated using jackknife resampling covariance matrix with 50 resampling samples (see Section
\ref{Spectroscopic Subsample}), please note that all the error bars in our graphs only
come from the jackknife resampling covariance matrix (there are some other errors like the error of the color
and stellar mass estimated by the Grid Method, and so on).

Note that the error bars for each subsample
are smallest on kiloparsec scales and become larger for both the
smaller and larger scales. On smaller scales this is due to shot
noise: the smaller the separations, the fewer the pair counts. On
scales larger than a few $h_{70}^{-1}\,\mathrm {Mpc}$, the errors grow in part
because our method becomes more and more
vulnerable to interlopers on larger scales where the clustering
power is weaker and background subtraction is more noisy.

Figures~\ref{fig:nwpredWITHred} through \ref{fig:nwpblueWITHblue}
show that $n_i \, w_{si}(r_p)$ scales something like $r_p$, but imperfectly.
We fit each set of $w_{si}(r_k)$ data with the smooth model:
\begin{equation} \label{eq:wrp0}
\widetilde{w}(r_p) = w_{0} [1 \, + \, \frac{r_p}{r_{c}}]^{\gamma} \, [\frac{r_p}{r_{c}}]^{-1}
\end{equation}
by minimizing
\begin{equation} \label{eq:chi2}
\chi_{si}^2 \equiv \sum_{k} \frac{(\rho_{i} \, \widetilde{w}(r_k) \, - \, \rho_{i} \, w_{si}(r_k))^2}{\rho_{i}^2 \, \sigma_{si}(r_k)^2}
\end{equation}
We fix $r_c$ at $r_c = 0.143 \, h_{70}^{-1}\,\mathrm {Mpc}$, and find the $\gamma$ and $\rho_i \, w_{0}$ that
minimizes $\chi_{si}^2$. Figures~\ref{fig:nwpredWITHred}
through \ref{fig:nwpblueWITHblue} show us these fits.

   \subsection{Three-dimensional statistics}

The smooth fit $w(r_p)$ to each projected correlation functions $w_{si}(r_p)$ can be
deprojected to get an estimate of the three-dimensional space correlation function $\xi(r)$ by
\begin{equation} \label{eq:nkxi}
\rho_{i} \, \xi(r) = -\frac {1}{\pi}\int_{r}^{\infty} \mathrm{d} r_p \frac{\mathrm{d} [\rho_{i} \, w_p(r_p)]}{\mathrm{d} r_p}(r_p^2-r^2)^{-1/2}
\end{equation}
\citep[\latin{e.g.},][]{davis83a}, where $\rho_i$ is a constant.

The mean total stellar mass $M_i^*$ of galaxies from a specific imaging
subsample $i$ within a given small three-dimensional separation $r_{close}$ around each galaxy from
spectroscopic subsample $s$ is:
\begin{equation} \label{eq:ni_temp}
M_i^* = \rho_i\int \mathrm{d} V_i \, [1 \, + \, \xi_{si}(r)] = 4 \, {\pi} \, \rho_i\int_{0}^{r_{close}} r^2 \, \mathrm{d} r \, [1 \, + \, \xi_{si}(r)]
\end{equation}
At small scales, $\xi_{si}(r) \gg 1$, so:
\begin{equation} \label{eq:Ni}
M_i^* \thickapprox 4 \, {\pi} \, \int_{0}^{r_{close}} r^2 \, \mathrm{d} r \, [\rho_i \, \xi_{si}(r)]
\end{equation}
we can see that the term in brackets in the approximate expression is the quantity we can de-project
from the two-dimensional projected cross-correlation function $\rho_i \, w_{si}(r_p)$. So $M_i^*$ is
directly related to the measurable quantities.

At very small scale ($r_{close} \ll r_c$):
\begin{equation} \label{eq:Mi0}
M_i^* \thickapprox 4 \, \rho_{i} \, w_{0} \, r_c \, r_{close}
\end{equation}

Similarly, the average number $N_i^*$ of galaxies from a specific imaging
subsample $i$ within a given small three-dimensional separation $r_{close}$ per galaxy from
spectroscopic subsample $s$ at very small scale ($0< r_{close}/r_c \ll 1$) is:
\begin{equation} \label{eq:nitemp}
N_i^* \thickapprox 4 \, n_{i} \, w_{0} \, r_c \, r_{close}
\end{equation}

   \subsection{Merger rate}

We can estimate the merger rate $\Gamma_i$ of galaxies from sample $i$ into galaxies from sample $s$ per
galaxy from the spectroscopic sample per unit time by:
\begin{equation} \label{eq:merger}
\Gamma_i = \frac{N_i^*}{t_{merge,i}}
\end{equation}
The mean fractional rate of growth of mass of galaxies from spectroscopic subsample $s$ from
accretion of galaxies from imaging subsample $i$ is:
\begin{equation} \label{eq:growth}
\left[\frac{\mathrm{d} \, \mathrm {ln} M_s}{\mathrm{d} t}\right]_i = \frac{1}{M_s}\left[\frac{\mathrm{d} M_s}{\mathrm{d} t}\right]_i \thickapprox \frac{M_i^*}{t_{merge,i} \, M_s}
\end{equation}

The shortest conceivable merger time $t_{merge,i}$ estimate (which
produces the largest conceivable estimate of the merger rate) is the
orbital time $t_{orbit}$. A more realistic estimate is a time $t_{DF,i}$
based on dynamical friction. But in principle all of these times can
be underestimates (and hence any merger rate based on close pairs can
be an overestimate) because there is undoubtedly a large number of
close pairs that will not merge on any short timescale. In what
follows, we present the (short) orbital time $t_{orbit}$ and (more realistic) dynamical friction
time $t_{DF,i}$ as two options, but then interpret our merger rate
estimates as upper limits.

All of these merger rate estimates depend, in principle, on the radius
$r_{close}$ inside of which we have counted close pairs. However,
from these fit lines over the range of interest ($r_p < 50 \, h_{70}^{-1}\,\mathrm {kpc}$) in Figures~\ref{fig:nwpredWITHred} through
\ref{fig:nwpblueWITHblue} we know,
$w_{si}(r_p)$ scales (something) like $r_p^{-1}$,
$\xi(r)$ scales (something) like $r^{-2}$, $N_i^*$ and $M_{i}^*$ scale (something)
like $r_{close}$. Similarly, both time-scales (orbital and
dynamical-friction) scale like $r_{close}$. For this reason, the inferred
merger and accretion rates (which are based on ratios of $N_i^*$ or $M_i^*$ with
the timescales) do \emph{not} depend strongly on the choice of $r_{close}$ at $r_{close} < 50 \, h_{70}^{-1}\,\mathrm {kpc}$.

The shortest conceivable mean merger time for a galaxy from the imaging sample merged into a more massive
red galaxy from the spectroscopic subsample is
\begin{equation} \label{eq:orbit}
t_{orbit} = \frac{2 \, \pi \, r_{close}}{1.5 \, \sigma_v}
\end{equation}
Where $1.5 \, \sigma_v$ is the average orbital velocity for a galaxy from the imaging sample
orbiting a more massive red galaxy from the spectroscopic sample with velocity
dispersion $\sigma_v$, see equation (\ref{eq:velocity_red}), and 1.5 just makes the $t_{orbit}$ conservative.
Similarly,
the shortest conceivable mean merger time for a galaxy from the imaging sample merged into a more massive
blue galaxy from the spectroscopic subsample is
\begin{equation} \label{eq:orbit_blue}
t_{orbit} = \frac{2 \, \pi \, r_{close}}{V_c}
\end{equation}
Where $V_c$ is the average orbital velocity for a galaxy from the imaging sample around a more massive blue galaxy from the spectroscopic sample, see equation (\ref{eq:velocity_red}).
The dashed lines in Figures~\ref{fig:growthred} and \ref{fig:growthblue} as a function of the
mass ratio $M_i/M_s$ show the unrealistically high merger-rate assumption of all possible
mergers taking place within one orbital time. The total fractional accretion rate is the area under each curve.

The Chandrasekhar approximation to dynamical friction is longer than
the dynamical time by a factor roughly equal to the ratio of the mass
of the heavier galaxy to the lighter one. This approximation may
actually be an underestimate of the total merger time found in
explicit $N$-body simulations \citep{Boylan-Kolchin07a}, which
serves to strengthen the interpretation of our merger rate estimates as
upper limits. For our case, the approximation becomes
\begin{equation} \label{eq:dyntime}
t_{DF,i} = t_{orbit} \frac{M_s}{M_i}
\end{equation}
where we assumed $M_s > M_i$.
The solid lines in Figures~\ref{fig:growthred} and \ref{fig:growthblue} as a function of the
mass ratio $M_i/M_s$ show the more realistic assumption of all possible
mergers taking place within one dynamical friction time, and the total fractional accretion rate is the area under the curves.

We integrate the mass accretion rate from mergers over all imaging subsamples and find the total
fractional accretion rate ($h_{70} \,\unit{Gyr}^{-1}$) of all the main galaxies. This is shown with dashed lines (for
$t_{orbit}$) and solid lines (for $t_{DF,i}$) in Figure~\ref{fig:growth1}.

Table \ref{tab:data1} shows the total fractional accretion rates of all the 20 spectroscopic subsamples for
both the orbital time-scale assumption and the dynamical friction time-scale assumption.

\section{Comparison to Previous Work}

\label{compare}

   \subsection{Comparison to previous clustering results}

We estimate not $w_p(r_p)$ but $n_i \, w_p(r_p)$. In order to estimate $w_p(r_p)$ and compare
with previous results, we need to estimate $n_i$:
\begin{equation} \label{eq:wpwithoutni}
w_p(r_p)=\frac{n_i \, w_p(r_p)}{n_i}
\end{equation}
It is difficult within the SDSS data to precisely
measure the real-space number densities for the imaging subsamples with stellar masses $M_i < 6 \times 10^9 h_{70}^{-2}\,\mathrm {M_{\odot}}$,
because there is only good spectroscopic information about bright members of the imaging sample.
However, for galaxies with stellar masses $M_i > 6 \times 10^9 \, h_{70}^{-2}\,\mathrm {M_{\odot}}$, the real-space number density $n_i$
for the imaging subsample $i$ is measurable:
\begin{equation} \label{eq:nitrue}
n_i=\frac{n_i^s}{\bar{f}}
\end{equation}
where $n_i^s$ is the average real-space number density for the corresponding spectroscopic subsample $s$
within its volume limit, and $\bar{f}$
is the mean fraction of Main targets for which a classification was obtained in the object's sector,
for NYU-VAGC spectroscopic sample $\bar{f} \thickapprox 0.91 $. We assume that $n_i$ is non-evolving
over the redshift range of interest.
We find the lower and upper redshift limits of the volume-limit for the
corresponding spectroscopic subsample $s$ in order to calculate $n_i^s$.

Galaxy clustering has been measured at intermediate and small scales
\citep{Zehavi05a, Zehavi10a, Masjedi06a, Chen09a, Li10a, White11a}.
Our results are consistent with the results of \citealt{Zehavi10a} (see Figure~\ref{fig:zehavi_wp}).
In order to generate our mass-threshold samples which are nearly the same as their luminosity-threshold samples,
we calculated $M^*$, the mean mass of the galaxies nearby their
luminosity-threshold $M_r$ as our mass-threshold, and
we cut $M_s > M^*$ and $M_i > M^*$ to generate the corresponding mass-threshold samples.
This turns our cross-correlation into an auto-correlation. In order to calculate $n_i^s$ for
the two subsamples with $M_r < -18.0$ and $M_r < -18.5$ in Figure~\ref{fig:zehavi_wp},
we use the peak real-space
number densities instead of the average real-space number densities, because there
are no obvious volume-limit in these two subsamples.

Similarly, we can measure $\rho_{i}$ and $w_{si}(r_p)$ separately
instead of $\rho_{i} \, w_{si}(r_p)$.

   \subsection{Comparison to previous merger rate results}

Please note that for all the merger time in this section, we use $t_{merge} = t_{DF,i}$.

Our results are consistent with recent measurements
of the merger rates based on counts of close pairs
\citep{Masjedi08a, Patton08a, kitzbichler08a, Bundy09a, De Propris10a, Robaina10a}.
The low dry merger rate an upper limit of $1.6$ $h_{70} \, \unit{percent \, per \, Gyr}$ for massive red
galaxies (red9) here is in good agreement
with a number of other estimates: at $z < 0.36$, \citealt{Masjedi08a} obtain an upper
limit of $1.2$ $h_{70} \, \unit{percent \, per \, Gyr}$ (converted from $1.7$ $h \, \unit{percent \, per \, Gyr}$) for the dry
merger rate of SDSS LRGs with $M_i < -22.75$;
at $0.45 < z < 0.65$, \citealt{De Propris10a} determine a $5 \sigma$ upper limit to the
dry merger rate of $0.7$ $h_{70} \, \unit{percent \, per \, Gyr}$ (converted from $1.0$ $h \, \unit{percent \, per \, Gyr}$)
for galaxies with $-23 < M(r)_{k+e,z=0.2} + 5 \log h < -21.5$
in the 2dF-SDSS LRG and QSO (2SLAQ) redshift survey.

\citealt{Robaina10a} find that the fraction of galaxies ($M > 5 \times 10^{10} h_{70}^{-2}\,\mathrm {M_{\odot}}$)
in pairs separated between 15 and $30 \, h_{70}^{-1}\,\mathrm {kpc}$ in 3D space is $f_{3Dpair}^{15 - 30 \, h_{70}^{-1}\,\mathrm {kpc}} = 0.01$
at $z = 0.1$, which is calculate by $(1.0 - 0.3) \times F(z)$, because they find that $30 \sim 40$ percent of galaxies in close pairs have
$r < 15 \, h_{70}^{-1}\,\mathrm {kpc}$ separations.
They also expect most of the mergers to be majors;
$\latin{i.e.}$, with mass rations between 1:1 and 1:4.
Our result of $f_{3Dpair}^{15 - 30 \, h_{70}^{-1}\,\mathrm {kpc}}$ is $0.02$ at $0.03 < z < 0.15$ with mass ratio between 1:1 and 1:4,
which is consistent with the result of \citealt{Robaina10a}.

With $t_{merge} = t_{DF,i}$ determined, we compute the volumetric merger rate (the number of
mergers per unit time per unit comoving volume) as a function of the
stellar mass of the primary or host galaxy. We call this the {\it
merger rate mass function} (merger rate MF) and denote it using the variable, $\Psi$.
Figure~\ref{fig:volume_bundy} shows the comparison of our major merger rate MF at $0.03 < z < 0.15$ with the major merger rate MF of \citealt{Bundy09a} at $0.4 < z < 0.7$ at mass ratio $m/M > 0.25$.
We also compute the $f_{pair}^{7.1 - 28.6 \, h_{70}^{-1}\,\mathrm {kpc}}$ from our $n_i \, w_p(r_p)$ and compare with
the results of \citealt{Bundy09a}, \citealt{Patton08a} and \citealt{kitzbichler08a} in Figure~\ref{fig:bundy}.
We can see that both of Figures~\ref{fig:volume_bundy} and ~\ref{fig:bundy} show that our
results are consistent with these previous results.

Our results are also consistent with recent merger rates predicted in theories of galaxy
formation in a cosmological context \citep{maller06a, Stewart09a}.
We estimate our merger rate $R_{mg}$ at a certain mass ratio $m/M$ by integrating
$\Gamma_i$ in equation (\ref{eq:merger}). We compare our $R_{mg}$
with the results of \citealt{maller06a} at mass ratio $m/M > 0.5$ (Figure~\ref{fig:maller}) and
the results of \citealt{Stewart09a} at mass ratio $m/M > 0.3$ and $m/M > 0.6$ (Figure~\ref{fig:stewart}).
We can see that both of Figures~\ref{fig:maller} and \ref{fig:stewart} show that our
results are consistent with these previous results.

\section{Discussion}

We find that under the more realistic dynamical friction assumption,
the total fractional growth rates onto red main galaxies are from $[1.25 \pm 0.04]$ to
$[1.82 \pm 0.02]$ $h_{70} \, \unit{percent \, per \, Gyr}$ depending on stellar mass, and those onto blue main galaxies
are from $[0.48 \pm 0.01]$ to $[0.56 \pm 0.03]$ $h_{70} \, \unit{percent \, per \, Gyr}$.
We find that at fixed stellar mass,
the total fractional accretion rates onto red galaxies is greater than that onto blue galaxies.
The total fractional accretion rate is a stronger function of primary mass for red galaxies
than that for blue galaxies.

Our results are consistent with recent results on the high redshift Universe which find
that the luminous red galaxies appear to grow in stellar mass over
time \citep{bell04a, willmer05a, blanton06a,
faber07a, conselice07a, Cimatti08a, vanDokkum08a, Saracco09a} if we assume that the accretion rate was
much higher in the past.
If we take our results at face value and assume
that the growth happens at a non-evolving rate, we expect the red galaxies to
grow by about $10$ to $15$ percent depending on stellar mass between redshift $z=1$ and $z=0.1$
(a period of $\approx 8\,h_{70}^{-1}\,\unit{Gyr}$).

According to Section \ref{compare},
our results are consistent with the previous clustering results of \citealt{Zehavi10a}.
Our estimated merger rates are consistent with the
merger rates estimated by counting of close pairs
\citep{Masjedi08a, Patton08a, kitzbichler08a, Bundy09a, De Propris10a, Robaina10a}.
Our estimated merger rates are also consistent with
the merger rates predicted in theories of galaxy
formation in a cosmological context \citep{maller06a, Stewart09a}.

However, not all merger studies find such low values.
The morphological derivations of the merger
fraction (\latin{e.g.}, \citealt{De Propris07a, conselice07a, lotz06a, Lotz08a, Conselice09a})
tend to find values of $f_{pair} \approx 0.1$ at $0.4 < z < 1.4$, about $2$
times higher than the results of our pair analysis (see Figure~\ref{fig:bundy}).
The discrepancy can be resolved easily if either
\textsl{(1)}~morphological signatures of merging last for many
dynamical times (\latin{e.g.}, tidal tails)
or \textsl{(2)}~the very minor mergers ($m/M < 0.25$) inflate the merger rates
(please note that our pair analysis in Figure~\ref{fig:bundy} are only estimated from major mergers at
mass ratio $m/M > 0.25$).

The mass growth rates shown in the
solid lines in Figure~\ref{fig:growth1} are upper limits on the true
fractional mass growth.
There are two reasons:
\textsl{(1)}~we assume $t_{merge,i} = t_{DF,i}$ for every pair.
However it is not true for close pairs in high velocity-dispersion
environments, and
the Chandrasekhar formula may be an overestimate \citep{Boylan-Kolchin07a} when the pairs are bound,
\textsl{(2)}~we assume that the stellar mass growth of the central galaxies from the
spectroscopic sample is equal to the stellar mass of the galaxy from the imaging sample, see equation (\ref{eq:growth}).
However \citealt{lin04a} found that up to $50$ percent of the stars in the galaxies from the imaging sample could be
stripped off before the merger with LRGs is complete.
So
our mass growth rate under dynamical friction assumption is an upper limit on the growth by merging.

We find that the accretion onto red and massive blue galaxies
is dominated by mergers with red companions, and that onto small-mass blue galaxies, red and
blue companions make comparable contributions, this is shown by Table \ref{tab:data1}
and Figures~\ref{fig:growthred} and \ref{fig:growthblue}.
So, most of the mass brought into red galaxies by merging is brought by red
companions or ``dry mergers'' \citep{bell06b, van Dokkum05a, Masjedi08a}, and most of it brought into
very massive galaxies (LRGs) is brought by galaxies
near (or above) $L^{\ast}$.

We find that all the contributions to growth decrease with decreasing
stellar mass at the small-mass end for all of 18
spectroscopic subsamples.
The contribution to growth decreases with
decreasing $M_i/M_s$ since $M_i/M_s < 0.4$.
For all 18 subsamples, the curves essentially decrease to zero
by $M_i/M_s < 0.01$ for $t = t_{orbit}$ and by $M_i/M_s < 0.1$ for $t = t_{DF,i}$, so calculation of the total amount of mass
brought in by merger activities does not require consideration of
galaxies from the imaging sample with $M_i/M_s < 0.01$.

The principal limitation of the imaging sample arises from the lack of
spectroscopic information on the galaxies from the imaging sample.
However, for galaxies from imaging sample with stellar masses $M_i > 6 \times 10^9 \, h_{70}^{-2}\,\mathrm {M_{\odot}}$,
we estimate the real-space number densities $n_i$ from equation (\ref{eq:wpwithoutni}).
So we measured $n_i$, $\rho_{i}$ and $w_{si}(r_p)$ separately.
But at stellar masses $M_i < 6 \times 10^9 h_{70}^{-2}\,\mathrm {M_{\odot}}$, it is impossible to precisely
measure the real-space number densities $n_i$,
so that we cannot disentangle the clustering power from the number density for
these small-mass galaxies from the imaging sample, and we only measure the products $n_i \, w_{si}(r_p)$ and $\rho_{i} \, w_{si}(r_p)$
but not either $n_i$, $\rho_{i}$ or $w_{si}(r_p)$ separately.

Figures~\ref{fig:nwpredWITHred} through \ref{fig:nwpblueWITHblue} show the
results of our measurements of $\rho_{i} \, w_{si}(r_p)$. The
sharp break at at $r_p \thickapprox 2.5 \, h_{70}^{-1}\,\mathrm {Mpc} $ and the less-sharp transition
at $r_p \thickapprox 0.43 \, h_{70}^{-1}\,\mathrm {Mpc} $,
which can also be observed in Figure~\ref{fig:zehavi_wp},
can be explained in the context of the ``halo occupation'' picture of galaxy clustering
\citep{peacock00a, scoccimarro01a, berlind02a, cooray02a, zheng05a, watson10a, watson11a},
and these two transitions are clearer in
Figure~\ref{fig:nwpredWITHred} of mergers between two red galaxies than that in the other
three figures.
The mergers at $r_p < 0.43 \, h_{70}^{-1}\,\mathrm {Mpc}$ are the one-halo mergers (both of the two merging galaxies
are inside one halo), and the mergers at $r_p > 2.5 \, h_{70}^{-1}\,\mathrm {Mpc}$ are the two-halo mergers
(the two merging galaxies are separately inside two nearby halos), and the mergers at $0.43 < r_p < 2.5 \, h_{70}^{-1}\,\mathrm {Mpc}$
are the mixed-halo mergers (some of the mergers are the one-halo mergers, the others are the two-halo mergers).
So at $r_p \thickapprox 0.43 \, h_{70}^{-1}\,\mathrm {Mpc}$ the mergers trasfer from the one-halo mergers to
the mixed-halo mergers, and at $r_p \thickapprox 2.5 \, h_{70}^{-1}\,\mathrm {Mpc}$ the mergers trasfer from
the mixed-halo mergers to the two-halo mergers.
These two transitions were also found in LRGs by \citealt{Masjedi08a}.

\clearpage


\section{Introduction}
A \emph{dominant eigenvalue} of a real square matrix is an eigenvalue of
maximum modulus. We call a square matrix \emph{Pisot} if it has
non-negative integer entries, its dominant eigenvalue is simple and all
eigenvalues different from the dominant one have absolute values less than one.
We prove that several monoids of non-negative matrices enjoy the
property of all being Pisot.

\medskip

Our first family of matrices is related to the so called \emph{fully subtractive} (multidimensional) continued fraction
algorithm. For an integer $d \geq 2$ we define
for each $k=1,\ldots,d$ the matrix $A_{FS,d}^{(k)}$ by
\[
(A_{FS,d}^{(k)})_{ij} = \left\{ \begin{array}{ll}
1 & \text{if $j=k$ or $i=j$}, \\
0 & \text{otherwise}
\end{array} \right.
\]
For $d=3$ this boils down to the three matrices
\[
A_{FS,3}^{(1)} = \begin{pmatrix}1&0&0\\1&1&0\\1&0&1\end{pmatrix},
\qquad
A_{FS,3}^{(2)} = \begin{pmatrix}1&1&0\\0&1&0\\0&1&1\end{pmatrix},
\qquad
A_{FS,3}^{(3)} = \begin{pmatrix}1&0&1\\0&1&1\\0&0&1\end{pmatrix}.
\]
All non-degenerate products of the matrices $A_{FS,d}^{(k)}$ satisfy the Pisot property.
\begin{theorem} \label{thm:fs}
Let $A = A_{FS,d}^{(i_1)} A_{FS,d}^{(i_2)} \ldots A_{FS,d}^{(i_n)}$ be a product of the fully subtractive matrices in dimension $d$.
Then the matrix $A$ is primitive if and only if all letters $\{1, \ldots, d\}$ appear in the sequence $(i_1, i_2, \ldots, i_n)$.
Moreover, if the matrix $A$ is primitive then it is Pisot.
\end{theorem}
Recall that a non-negative square matrix $A$ is primitive if there exists a positive integer $n$ so that $A^n$ has all its
entries positive. The case $d=3$ of Theorem~\ref{thm:fs} was proved in~\cite{ArnouxIto01}. The authors used an induction
on characteristic polynomials and our approach is radically different.

\medskip

The same result holds for another set of $3 \times 3$ matrices related to the Brun multidimensional
continued fractions. Let
\[
A_{Br}^{(1)} = \begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix},
\qquad
A_{Br}^{(2)} = \begin{pmatrix}1&1&0\\1&0&0\\0&0&1\end{pmatrix},
\qquad
A_{Br}^{(3)} = \begin{pmatrix}1&0&1\\1&0&0\\0&1&0\end{pmatrix}.
\]
\begin{theorem} \label{thm:brun}
Let $A = A_{Br}^{(i_1)} A_{Br}^{(i_2)} \ldots A_{Br}^{(i_n)}$ be a product of the $3 \times 3$ Brun matrices.
Then, $B$ is primitive if and only if the matrix $B^{(3)}$ appears in the product.
Moreover, if $B$ is primitive then it is Pisot.
\end{theorem}
This result was already known since the work of Brun~\cite{Brun57}.

\medskip

The proofs of Theorem~\ref{thm:fs} and~\ref{thm:brun} are elementary and uses the following inequality.
Given a non-negative primitive $d \times d$ matrix $A$ and its Perron-Frobenius eigenvector $v \in \mathbb{R}^d_+$ the absolute value $\lambda_2$
of its second largest eigenvalue satisfies
\[
\lambda_2 \leq \sup_{x \in v^\perp \backslash \{0\}} \frac{\|A x\|}{\|x\|}.
\]
where $\|.\|$ is any norm on $\mathbb{R}^d$. In our proof, the information we have on the localization of the Perron-Frobenius eigenvector
comes from the dynamical systems induced by the matrices; in other words the fully subtractive and Brun continued fraction algorithms.

\medskip

From a diophantine approximation point of view,
the Pisot property is particularly interesting because it provides the so called
exponential convergence of the continued fraction
expansion for almost every vectors (see~\cite{Lagarias}). We show
that the above results naturally extends to this situation in Section~\ref{sec:lyapunov}.

\medskip

Beyond continued fractions, Pisot matrices are of special interest in
substitutive dynamical systems. More precisely, replacing matrices with so
called substitutions, the associated dynamical systems admit $d-1$ eigenvalues
and in many cases the dynamical system can be proved to have purely discrete
spectrum (see~\cite{Pytheas} Chapter~7). This was the main motivation for the
study of the fully subtractive matrices in~\cite{ArnouxIto01}.

\section{Fully subtractive and Brun continued fractions} \label{sec:cf}
Let us consider a finite or countable set $\mathcal{A}$ that we call alphabet and for each $i \in \mathcal{A}$
a matrix $A^{(i)} \in \operatorname{SL}(d,\mathbb{Z})$. We already saw two examples of this with
the fully subtractive algorithm where $\mathcal{A}_{FS,d} = \{1,2,\ldots,d\}$ and for Brun
algorithm where $\mathcal{A}_{Br} = \{1,2,3\}$.

To the data $(\mathcal{A}, (A^{(i)})_{i \in \mathcal{A}})$ we associate the set of infinite words $\Delta = \mathcal{A}^\mathbb{N}$,
the shift map $T: \Delta \rightarrow \Delta$ and a cocycle
\[
\forall x \in \Delta, \forall n \geq 0, \quad A_n(x) = A^{(x_0)} A^{(x_1)} \ldots A^{(x_{n-1})}.
\]
The maps $A_n: \Delta \rightarrow \operatorname{SL}(d,\mathbb{Z})$ satisfy the so called \emph{cocycle property}:
$A_{m+n}(x) = A_m(x) A_n(T^m x)$.

\begin{definition}
Let $(A^{(i)})_{i \in \mathcal{A}}$ be a set of matrices in $\operatorname{SL}(d,\mathbb{Z})$ where $\mathcal{A}$ is a finite or countable alphabet.
We say that a set $D \subset \mathbb{P}(\mathbb{R}^d)$ is \emph{adapted} to these matrices if it is non-empty, it is the
closure of its interior and for all $i \in \mathcal{A}$ we have $A^{(i)} D \subset D$.
\end{definition}
For example $PP(\mathbb{R}^d)$ is always adapted. But we will be interested in the somewhat
smallest adapted set in order to localize the dominant eigenvector.

Given $(A^{(i)})_{i \in \mathcal{A}}$ and $D \subset \mathbb{P}(\mathbb{R}^d)$ adapted, we define $D^{(i)} = A^{(i)} D$
and more generally for a finite word $w = i_0 i_1 \ldots i_{n-1}$ we define
$D^{(w)} = A^{(i_0)} A^{(i_1)} \ldots A^{(i_{n-1})} D$. Note that for any $w$ the set
$D^{(w)}$ is \emph{not} empty. Given an infinite word $x = x_0 x_1 \ldots \in \Delta$ we also
set $D_n(x) = A_n(x) D$ and $D_\infty(x) = \bigcap_{n \geq 0} D_n(x)$.

Let $A \in \operatorname{SL}(d,\mathbb{Z})$ and let $\lambda$ be its spectral radius. Consider its
Jordan decomposition over $\mathbb{C}$ and the Jordan blocks
associated with an eigenvalue of modulus $\lambda$ and being of maximal
dimension. To each of these maximal Jordan block is associated exactly one
eigenvector $v_i$. The \emph{dominant eigenspace} of $A$ is $\mathbb{R}^d \cap (\mathbb{C}
v_1 \oplus \mathbb{C} v_2 \oplus \ldots \mathbb{C} v_r)$. We have the following elementary
result.
\begin{lemma} \label{lem:dominant_eigenspace}
Let $(A^{(i)})_{i \in \mathcal{A}}$ be a finite or countable set of matrices in $\operatorname{SL}(d,\mathbb{Z})$ and
let $D$ be adapted. Let $x = x_0 x_1 \ldots \in \Delta$ be an infinite word over $\mathcal{A}$.
Then for any $n$, the set $D_n(x)$ contains a basis of the dominant
eigenspace of $A_n(x)$.
\end{lemma}
We omit the proof that only uses the fact that the maximum growth of $\|A^n u\|$ is
$\lambda^n n^k$ where $k$ is the  maximal dimension of a Jordan block
associated with an eigenvalue of maximal modulus of $A$.

\medskip

Let
\[
D_{FS,d} = \{(x_1,\ldots,x_d) \in \mathbb{P}(\mathbb{R}_+^d): \forall i,j,k \quad x_i < x_j + x_k \}
\]
and
\[
D_{Br} = \{(x,y,z) \in \mathbb{P}(\mathbb{R}_+^3):\ x > y > z \}.
\]
Then it is easily seen that $D_{FS,d}$ is adapted for the fully subtractive matrices in dimension $d$
and $D_{Br}$ is adapted for the Brun matrices. In figures~\ref{fig:fs} and~\ref{fig:brun} one can see the projective picture
of the domains $D^{(1)}$, $D^{(2)}$ and $D^{(3)}$. Note that in these cases, the domains $D^{(i)}$ are
disjoint but that it is not a requirement in our definition. Moreover, one can see that in the Brun
case the $D^{(i)}$ form a partition while it is not the case for the fully subtractive.
\begin{figure}[!ht]
\centering
\begin{minipage}{0.4\textwidth}
\begin{gather*}
A_{FS,3}^{(1)}=\begin{pmatrix}1&0&0\\1&1&0\\1&0&1\end{pmatrix}
\qquad
A_{FS,3}^{(2)}=\begin{pmatrix}1&1&0\\0&1&0\\0&1&1\end{pmatrix}
\\
A_{FS,3}^{(3)}=\begin{pmatrix}1&0&1\\0&1&1\\0&0&1\end{pmatrix}.
\end{gather*}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\centering \includegraphics{FullySubtractiveDomain.pdf}
\end{minipage}
\caption{Fully subtractive partition of the domains with $d=3$.}
\label{fig:fs}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\begin{minipage}{0.4\textwidth}
\begin{gather*}
A_{Br}^{(1)} = \begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix}
\qquad
A_{Br}^{(2)} = \begin{pmatrix}1&1&0\\1&0&0\\0&0&1\end{pmatrix}
\\
A_{Br}^{(3)} = \begin{pmatrix}1&0&1\\1&0&0\\0&1&0\end{pmatrix}.
\end{gather*}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\begin{center}
\includegraphics{BrunDomain.pdf}
\end{center}
\end{minipage}
\end{center}
\caption{The matrices and domains for the Brun algorithm.}
\label{fig:brun}
\end{figure}

If the $D^{(i)}$ are disjoint one can define a continued fraction algorithm
as follows. One defines a partial map $f: D \dashrightarrow D$ by setting
$fx = (A^{(i)})^{-1} x$ on $D^{(i)}$.

One can compute that for Brun one has
\[
f_{Br}(x) = \operatorname{sort}(x-y,y,z)
\]
where $\operatorname{sort}: \mathbb{P}(\mathbb{R}_+^3) \rightarrow D$ is the map which permutes the coordinates in order to sort them.
While for the fully subtractive one has
\[
f_{FS,d}(x) = (x_1-x_i, x_2-x_i, \ldots, x_{i-1}-x_i, x_i, x_{i+1}-x_i, \ldots, x_d - x_i)
\qquad
\text{if $x_i = \min(x_1, \ldots, x_d)$}.
\]

\section{Strategy}
The proofs of Theorems~\ref{thm:fs} and~\ref{thm:brun} follow a general strategy that we describe now.
We let $\|.\|$ be the $L^\infty$ norm on $\mathbb{R}^d$ and the associated operator norm on matrices. That is
for a vector $v$ and a matrix $A$
\[
\|v\| = \max (|v_1|, \ldots, |v_d|)
\quad \text{and} \quad
\|A\| = \max_{i=1,\ldots,d} \sum_{j=1,\ldots,d} |a_{ij}|.
\]
In this section the norm used on $\mathbb{R}^d$ has no importance. But it turns out
that, to apply the results to continued fraction algorithms, the most convenient one
was always the $L^\infty$ norm.

To a non-zero vector $v$ in $\mathbb{R}^d$, we associate its dual hyperplane $H_v = \{z \in \mathbb{R}^d; \langle v,z \rangle = 0\}$.
Given a non zero vector $v$ in $\mathbb{R}^d$ we define the following semi-norm on $d \times d$ matrices
\[
\|B\|_v = \sup_{z \in H_v \backslash \{0\}} \frac{\|B z\|}{\|z\|} = \max_{\stackrel{\|z\| \leq 1}{z \in H_v}} \|B z\|.
\]
More generally, if $\Lambda \subset \mathbb{R}^d$ is a cone, we define
\[
\|B\|_\Lambda = \sup_{v \in \mathbb{P}(\Lambda)} \|B\|_v.
\]

Let $(A^{(i)})_{i \in \mathcal{A}}$ be a finite or countable set of matrices as in Section~\ref{sec:cf}.
Let also $\Delta = \mathcal{A}^\mathbb{N}$ and $D \subset \mathbb{P}(\mathbb{R}^d)$ be adapted.
Recall that $D_n(x) = A_n(x) D = D^{(x_0 x_1 \ldots x_{n-1})}$ (in particular, $D_0(x) = D$ and $D_1(x) = D^{(x_0)}$)
and that $D_\infty(x) = \bigcap_{n \geq 0} D_n(x)$.
\begin{lemma}  \label{lem:strategy}
Let $(A^{(i)})_{i \in \mathcal{A}}$ be a finite or countable set of matrices in $\operatorname{SL}(d,\mathbb{Z})$ and let
$D \subset \mathbb{P}(\mathbb{R}^d)$ be adapted. Let $B^{(i)}$ (respectively
$B_n(x)$) denote the transposed of $A^{(i)}$ (resp. $A_n(x)$).
If for all $i \in \mathcal{A}$ we have
\[
\left\| B^{(i)} \right\|_{D^{(i)}} \leq 1.
\]
Then for any point $x = x_0 x_1 \ldots \in \Delta$ we have
\[
\| B_n(x) \|_{D_n(x)} \leq 1.
\]
In particular, if $x = (x_0 x_1 \ldots x_{p-1})^\infty$ is periodic the matrix
$A_p(x) = A^{(x_0)} A^{(x_1)} \ldots A^{(x_{p-1})}$ has at most one eigenvalue
greater than one in absolute value.
\end{lemma}

\begin{proof}
Let $B$ be the transposed cocycle.
The hypothesis is just the case $n=1$. Assume that this inequality holds for $n$.
By definition $A_{n+1}(x) = A_1(x) A_n(Tx)$ and $D_{n+1}(x) = A_1(x) D_n(Tx)$.
Hence $v \in D_n(Tx)$ if and only if $A_1(x) v \in D_{n+1}(x)$.
Let us choose $v \in D_n(Tx)$, then
\[
\| B_{n+1}(x) \|_{A_1(x)v} = \| B_n(Tx) B_1(x) \|_{A_1(x)v} \leq \|B_n(Tx)\|_v \cdot \|B_1(x)\|_{A_1(x)v}.
\]

Now, let $x = (x_0 x_1 \ldots x_{p-1})^\infty$ be a periodic point. Because
$D_{p-1}(x) \supset D_\infty(x)$, the matrix $B_p(x)$ satisifies
\[
\| B_p(x) \|_{D_\infty(x)} \leq 1.
\]
By Lemma~\ref{lem:dominant_eigenspace} a basis of the dominant eigenspace of $B_p$ belongs to
$D_\infty(x)$. Consequently, the union of the orthogonals of $D_\infty(x)$ contains
all eigenspaces corresponding to the non-dominant eigenvalues. From the above inequality,
we deduce that the absolute value of all eigenvalues different from the first one are bounded by $1$.
\end{proof}

\section{Pisot property for Arnoux-Rauzy matrices} \label{sec:fs}
In this section we prove Theorem~\ref{thm:fs}. Let $d \geq 2$ be an integer
and let $e_1$, $e_2$, \ldots $e_d$ be the canonical
basis of $\mathbb{R}^d$. Let $e = e_1 + e_2 + \ldots + e_d$ and for $i=1,\ldots,k$ let $f_i = e - e_i$.
The domain $D_{FS,d}$ is the convex hull of the rays vectors $\mathbb{R}_+ f_i$.

Let as usual $B^{(w)} = (A^{(w)})^*$ and $B_n(x) = (A_n(x))^*$.
We claim that we even have a stronger property than what is required in Lemma~\ref{lem:strategy}
\[
\forall i=1,\ldots,d, \qquad \| B^{(i)} \|_D \leq 1.
\]
Let us prove this claim.
Let $v \in D_\infty$, then we may write $v = \mu_1 f_1 + \mu_2 f_2 + \ldots + \mu_d f_d$
for some non-negative numbers $\mu_i$ that satisfy $\mu_1 + \mu_2 + \ldots + \mu_d = 1$.
We hence have $v = e - \sum \mu_i e_i$ and
\[
H_v = \{z \in \mathbb{R}^d; \langle z,v \rangle = 0\} = \{z \in \mathbb{R}^d; \sum z_j = \sum \mu_j z_j\}.
\]
Given $z \in H_v$ we have $B^{(i)} z = (z_1, \ldots, z_{i-1}, \sum \mu_z z_j, z_{i+1}, \ldots, z_d)$,
In other words, $B^{(i)}$ acts on $H_v$ as a stochastic matrix $P(i,v)$ which is the identity except
its $i$-th row which is $(\mu_1,\mu_2,\ldots,\mu_d)$.
In particular, $\|B^{(i)}\|_v \leq 1$.

\medskip

Now for a given finite product $A = A^{(i_0)}A^{(i_1)} \ldots A^{(i_{p-1})}$ if one of the letter
$\{1,\ldots,d\}$ is missing in the sequence $(i_0, i_1, \ldots, i_{p-1})$ then $A e_i = e_i$ and
so the matrix is not primitive. On the other hand, if all letters appear it is easy to see
that all entries in $A$ are positive.

\medskip

Now let $x = (x_0 x_1 \ldots x_{p-1})^\infty$ be a periodic point that contains all letters from $\mathcal{A}$.
Because of positivity, all the orbits is contained in the interior of $D$ and the dominant eigenvalue
is simple. Let $v_0 \in D$ be a dominant eigenvector and let $v_n = A_n(x) v_0$. Because all $(v_n)$
belongs to the interior of $D$ the coefficients $\mu_1$, $\mu_2$, \ldots, $\mu_d$ that appear in
the stochastic matrices $P(x_i, v_i)$ are all positive. Now, the product
$P(x_{p-1}, v_{p-1}) \ldots P(x_1, v_1) P(x_0, v_0)$ is a stochastic matrix with all its entries positive.
Hence, its second eigenvalue, which is also the second eigenvalue of $A_p(x)$, is less than $1$ in absolute
value.

\section{Pisot property for Brun algorithm (in dimension 3)}
We now turn to the proof of Theorem~\ref{thm:brun}.
Let $\mathcal{A} = \{1,2,3\}$ and $A^{(1)}$, $A^{(2)}$, $A^{(3)}$ be the matrix of the Brun algorithm.
We let $\Delta = \mathcal{A}^\mathbb{N}$ and denote by $A$ and $B$ respectively the cocycle and the transposed
cocycle.
We claim that, as in the case of the fully subtractive, we have the stronger property that
\[
\forall i \in \{1,2,3\},\quad  \|B^{(i)}\|_{D^{(i)}} \leq 1.
\]
We only need to consider the matrix $\displaystyle B^{(1)} =
\begin{pmatrix}1&0&0\\1&1&0\\0&0&1\end{pmatrix}$ since the other two are
obtained by multiplying by a permutation matrix which will not change the $L^\infty$-norm.

Let $v = \mu_1 (1:0:0) + \mu_2 (1:1:0) + \mu_3 (1:1:1) \in D$ for some $\mu_1,\mu_2,\mu_3$ such that $\mu_1+\mu_2+\mu_3=1$
and $H_v = \{z \in \mathbb{R}^d; z_1 + z_2 = \mu_1 z_2 - \mu_3 z_3 \}$.
Now, for any $z \in H_v$ we have $B^{(1)} (z_1,z_2,z_3) = (z_1,z_1+z_2,z_3) = (z_1, \mu_1 z_2 - \mu_3 z_3, z_3)$.
In other words $\|B^{(1)} (z_1,z_2,z_3)\|_1 \leq \|(z_1,z_2,z_3)\|$.

\medskip

Now, given a product $A = A_Br^{(i_1)} \ldots A_Br^{(i_n)}$ it is easy to see that if $3$ does not appear in
the sequence $(i_1, i_2, \ldots, i_n)$ then $A e_3 = e_3$ and hence the matrix $A$ can not be irreducible.
Conversly, if $3$ appears then $A^3$ is easily seen to be positive.

\medskip

Now consider the matrix $P$ built from the begining of the proof. As in the case of the fully subtractive
algorithm for a primitive product we got that the $\mu_i$ are all positive. Given a product $A$ where
the matrix $A_{Br}^{(3)}$ appears, the matrix $A^3$ is then such that all rows are such that sum of their
absolute values is strictly less than one. In other words $\|A^3\| < 1$.

\section{Lyapunov exponents} \label{sec:lyapunov}
Let $(A^{(i)})_{i \in \mathcal{A}}$ be a finite or countable set of matrices.
Let $\Delta$, $T$, $A$, $B$ denote as before the infinite words, the shift map
the cocycle and the transposed cocycle. Let also $D$ be adapated to these matrices.

The asymptotic of the cocycle (or the transposed cocycle) are studied through \emph{Lyapunov
exponents}. Given a $T$-invariant ergodic probability measure $\mu$ on $\Delta$, we associate
the real numbers $\gamma^\mu_1 \geq \gamma^\mu_2 \geq \ldots \geq \gamma^\mu_d$ defined by
\[
\forall k \in \{1,2,\ldots,d\}, \quad
\gamma_1 + \gamma_2 + \ldots + \gamma_k = \lim_{n \to \infty} \int_\Delta \frac{\log \|\wedge^k A_n(x)\|}{n} d\mu(x).
\]
In order to be well defined we assume that
\begin{equation} \label{eq:integrability}
\int_\Delta \max \left(\log \|A_1(x)\|, \log \|A_1(x)^{-1}\|\right) d\mu(x) < \infty
\end{equation}
and we refer to this condition as the \emph{$\log$-integrability} of the cocycle.
If the alphabet $\mathcal{A}$ is finite the cocycle is automatically $\log$-integrable.
If $x$ is a periodic point of $T$ and $\mu = (\delta_{x} + \delta_{Tx} + \ldots + \delta_{T^{n-1}x}) / n$ is
the sum of Dirac masses distribued along its orbit, then the associated Lyapunov exponents are the logarithms of
the absolute values of eigenvalues of $A_n(x)$ where $n$ is the period of $x$.
In that sense, Lyapunov exponents generalize eigenvalues.

Given a measure $\mu$ for which the cocycle is $\log$-integrable, we say that $(\Delta,T,A,\mu)$ has \emph{Pisot spectrum}
if the associated Lyapunov exponents satisfy $\gamma_1^\mu > 0 > \gamma_2^\mu$. This property is related to the
strong convergence of higher dimensional continued fraction algorithm~\cite{Lagarias}.

Now we restate Lemma~\ref{lem:strategy} in a more dynamical context.
\begin{lemma} \label{lem:lyapunov}
Let $(A^{(i)})_{i \in \mathcal{A}}$ be a finite or countable set of non-negative matrices in $\operatorname{SL}(d,\mathbb{Z})$.
Let $(\Delta,T,A,B)$ be the associated full shift with its cocycle and its transposed cocycle.
Let also $D$ be adapated.
Assume that
\[
\forall i \in \mathcal{A}, \quad  \left\| \left(A^{(i)}\right) \right\|_{D^{(i)}} \leq 1.
\]
Let $\mu$ be a $T$-invariant and ergodic measure on $D$ so that
\begin{itemize}
\item the cocycle $A_n$ is $\log$-integrable,
\item there exists a cylinder $[w]$ such that $\mu([w]) > 0$, $A^{(w)}$ is positive and $\|A^{(w)}\|_{D^{(w)}} < 1$.
\end{itemize}
Then two first Lyapunov exponents of the cocycle $A_n$ for the measure $\mu$ satisfies $\gamma_1^\mu > 0 > \gamma_2^\mu$.
\end{lemma}

\begin{proof}
Let us first prove that $\gamma_1 > 0$.

Now, by definition, for $\mu$-almost every $x$
\[
\gamma_1 = \lim_{n \to \infty} \frac{\log \|A_n(x)\|}{n}
\]
Let $m=|w|$ be the length of $w$ and consider the position which are multiple of $m$.
For a $\mu$-generic $x$ we have by Birkhoff theorem that
\[
\lim_{n \to \infty} \frac{ \# \{i \leq n:\ T^i(x) \in [w]\}} {n} = \mu([w])
\]
In other words, given sequence of length $n$ large enough we can find a linear number of
disjoint occurrences of $w$ (up to a sublinear error). Let $k_n$ be the number of these occurrences, then
necessarily each entry of $A_n(x)$ is larger than the corresponding one in
$C^{k_n}$ where $C$ is the matrix which contains a $1$ in every position.  In
particular $\gamma_1 > 0$.

From the existence of $w$ it also follows that for a $\mu$-generic $x$ the cone $D_\infty(x)$ is reduced to
a line contained in the interior of $\mathbb{R}_+^d$. We can hence define $\mu$-almost everywhere a function
$v: \Delta \rightarrow \mathbb{R}_+^d$ by $D_\infty(x) = \mathbb{R}_+ v(x)$ and $\|v(x)\| = 1$.
We then have the following formulas which holds for $\mu$-almost every $x$
\[
\gamma_1 = \lim_{n \to \infty} \frac{- \log \|A_n(x)^{-1} v(x)\|}{n} d\mu(x)
\quad \text{and} \quad
\gamma_2 = \lim_{n \to \infty} \frac{\log \|B_n(x)\|_{v(x)}}{n} d\mu(x).
\]
It is then easy to derive the estimate for $\gamma_2$. The map $x \mapsto v(x)$ and the
dual hyperplanes $H_{v(x)}$ satisfy the following covariance properties
\[
D_\infty(Tx) = A(x)^{-1} D_\infty(x)
\quad \text{and} \quad
H_{A^{-1} v} = A^* H_v.
\]
Hence as in the proof of Lemma~\ref{lem:strategy}, we deduce that for all $v \in D_\infty(x)$
\[
\|B_{m+n}(x)\|_{v(x)} \leq \|B_{m}(x)\|_{v(x)} \|B_{n}(T^mx)\|_{v(T^m x)}
\]
In particular, for $\mu$-almost every $x \in [w]$, any $n \geq |w|$ we get that $\|B_n(x)\|_{v(x)} < 1$.
Let $\delta = \|B^{(w)}\|_{D^{(w)}} < 1$.
Using the same argument as in the estimation of $\gamma_1$ we get that
\[
\gamma_2 \leq \liminf_{n \to \infty} \frac{k_n \log \delta}{n}.
\]
And the above limit is strictly negative.
\end{proof}