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\begin{document}
\begin{sf}
\title{
\ { \sf ON THE $1/N$ CORRECTIONS TO THE GREEN FUNCTIONS OF RANDOM MATRICES WITH
INDEPENDENT ENTRIES}
}
\author{\sf A.Khorunzhy, B.Khoruzhenko, and L.Pastur
\\Mathematical Division, Institute for Low Temperature Physics, \\47 Lenin Ave, 310164,
Kharkiv, Ukraine}
\date{ }
\maketitle
\begin{abstract}
{\sf We propose a general approach to the construction of $1/N$ corrections to the Green function
$G_N(z)$ of the
ensembles of random real symmetric and Hermitean $N \times N$ matrices with independent entries
$H_{k,l}$. By this approach we study the correlation function $C_N(z_1,z_2)$
of normalized trace $N^{-1}\mbox{Tr}\;G_N$ assuming that the average of $|H_{k,l}|^5$
is bounded. We found that to the leading order $C_N(z_1,z_2)=N^{-2}F(z_1,z_2)$, where
$F(z_1,z_2)$ depends only on the second and the fourth moments of $H_{k,l}$.
For the correlation function of the density of energy levels we obtain an expression which,
in the scaling limit depends only on the second moment of
$H_{k,l}$. This can be viewed as a support to the universality conjecture of
random matrix theory}
\end{abstract}
1. Random $N\times N$ matrices with independent entries were introduced by
E.Wigner \cite{Wigner1958}.
The majority of rigorous results for these
matrices concerns the proof of convergence of the density $\rho_N (E)$
of their eigenvalues to the celebrated Wigner semicircle law and its
generalization known as the deformed semicircle law
\cite{Pastur1973,Brody_et_al1981,Girko1988}. These nonrandom limiting
eigenvalue distributions are completely determined by two first moments
of the probability distribution of random matrix entries.
Much less is
known on the large-$N$ corrections, in particular on their dependence
on the probability distribution of entries. The aim of this letter is
to present a rigorous approach to the systematic construction of the large-$N$
corrections for moments of the Green functions of respective random matrices.
\bigskip
2. We consider an ensemble of real symmetric $N \times N$ random matrices
which, in particular case of the Gaussian distributed entries, possesses the
orthogonal invariance property (respective ensemble is known as the
Gaussian orthogonal ensemble (GOE) and is an archetype model in the field
\cite{Mehta1991}). Thus, our random matrices $H=\{H_{k,l}\}_{k,l=1}^N$
are specified by relations
\begin{eqnarray}
H_{k,l}=H_{l,k}={ \Big( \frac {1+\delta_{k,l}}{N} \Big)}^{1/2} \,
W_{k,l}, \qquad k,l=1,\ldots, N.
\label{1}
\end{eqnarray}
Here $W_{k,l}$, $k\le l$, are independent random variables such that:
$\mbox{i)} \; \o {W_{k,l}}=0; \; \mbox{ii)} \; \o {W_{k,l}^2}\equiv v^2; \;
\mbox{iii)} \; \o {W_{k,l}^4}-3{ (\o {W_{k,l}^2}})^2\equiv \sigma; \;
\mbox{iv)} \; \max_{k\le l}\, \o {|W_{k,l}|^5} \leq C,$
where the bar denotes averaging over respective probability distribution
and $v, \; \sigma, \; \mbox{and} \; C$ are $N$-independent. Thus
$W_{k,l}$'s may have different probability distributions for different pairs
$(k,l)$ but their second and fourth moments have to be the same.
Denote by $G_N(z)=\{G_{i,k}(z)\}_{i,k=1}^N$ the Green function
$G_N(z)=(H_N-z)^{-1}, \; \mbox{Im} \; z \neq 0$, of our matrices and by
$\la G_N(z) \ra $ its normalized trace $\la G_N(z) \ra = 1/N \sum_{k=1}^N \,
G_{k,k}(z)$. It is known \cite{Girko1988,Khorunzhy_et_al1992}
that the moments of
$\la G_N(z) \ra $ factorize. Namely, if
\begin{eqnarray}
m_N^{(p)}(z_1,\ldots ,z_p)= \o {\prod_{i=1}^N \, \la G_N(z_i) \ra} ,
\label{2}
\end{eqnarray}
then
\begin{eqnarray}
m_N^{(p)}(z_1,...,z_p)=\prod^p_{i=1} \, m_N^{(1)}(z_i) + O(1/N^2).
\label{3}
\end{eqnarray}
In particular, the correlation function
\begin{eqnarray}
C_N(z_1,z_2)=\o {\la G_N(z_1) \ra \la G_N(z_2) \ra} - \o {\la G_N(z_1) \ra}
\cdot \o {\la G_N(z_2) \ra}
\label{4}
\end{eqnarray}
for $\mbox{Im} \; z_1, \, \mbox{Im} \; z_2 \neq 0$ is of the order $1/N^2$
as $N \to \infty$.
Our aim is to construct the expansion of $m_N^{(p)}(z_1,...,z_p)$ in powers of
$1/N$. We outline the main idea and find, as an example, the explicit form
of $1/N^2$ correction to $C_N(z_1,z_2)$. We also discuss some implications
of our results.
\bigskip
3. Our approach is an adaptation and extension of that proposed in
\cite{KhorunzhyPastur1993} to study spectral characteristics of a
certain class of random finite-difference operators of the order
$R$, acting in $l^2({\bf Z}^d)\otimes {\bf C}^n$ in the limiting cases
when one of the parameters $R, \, n$, or $d$ tends to infinity. The main idea
is to derive certain identities for the moments
$m_N^{(p)}(z_1,...,z_p)$ and then, treating the whole set of these relations
as an equation in an appropriate linear space, to compute the moments in each
order of $1/N$ by iterating this equation.
To derive the relations let us rewrite (\ref{2}) as
$1/N\sum_{k=1}^N \, \o {
G_{k,k}(z_1) \, \prod_{i=2}^p \, \la G_N(z_i) \ra
}$
and replace $G_{k,k}(z_1)$ by the r.~h.~s. of the resolvent identity
$G_{k,k}(z_1)=-1/z_1+1/z_1\,\sum_{l=1}^N \, H_{k,l}G_{l,k}(z_1)$. We obtain
\begin{eqnarray}
m_N^{(p)}(z_1,...z_p)= -\frac {1}{z_1} m^{(p-1)}(z_2,...z_p) +
\sum_{k,l=1}^N \; \o {
H_{k,l}G_{l,k}(z_1)\prod^p_{i=2} \; \la G_N(z_i)\ra
},
\label{5}
\end{eqnarray}
\noindent where for $p=0$ we set $m_N^{(0)}\equiv 1$. Now we average in the second
term of the r.h.s. of (\ref{5}) over the random variable $H_{k,l}$ by using
another resolvent identity
\begin{eqnarray}
G_{m,n}=\hat G_{m,n} -
\left[ \hat G_{m,k}G_{l,n}-\hat G_{m,l}G_{k,n}\right] H_{k,l}
(1+\delta_{k,l})/2,
\label{6}
\end{eqnarray}
where $\hat G = G\left|_{H_{k,l}=0} \right.$. Iterating (\ref{5}) several times
we represent any matrix element of $G_N(z)$ as a sum of powers of $H_{k,l}$
multiplied by matrix elements of $\hat G_N$ and of a power of $H_{k,l}$,
say $H_{k,l}^a$, multiplied by a sum of matrix elements of the both Green functions
$G_N$ and $\hat G_N$ ($a$ is determined by the number of iterations).
The inequality
\begin{eqnarray}
| G_{k,m}(z) | \le \left\| G_N \right\| \le {|\mbox {Im}\,z|}^{-1},
\label{7}
\end{eqnarray}
and analogous inequality for $\hat G_N$ allows us to
estimate the average of the latter by $\o{|H_{k,l}|^a}$
(which is proportional to $1/N^{a/2}$ according to (\ref{1})) multiplied
by a power of $|\mbox{Im}\;z|^{-1}$ and by some absolute constant.
Since $\hat G_N$ and $H_{k,l}$ are independent,
all other terms in this representation will have the form of moments of matrix
elements of $\hat G_N$ multiplied by moments of $H_{k,l}$ whose order is
smaller than $a$. At last, we return back from $\hat G_N$ to $G_N$ by using the
identity that differs from (\ref{6}) by interchanging $G_N$ and $\hat G_N$ and
replacing $H_{k,l}$ by $-H_{k,l}$. We obtain the identity
\begin{eqnarray}
m_N^{(p)}=(Am_N)^{(p)}+f_N^{(p)}.
\label{8}
\end{eqnarray}
Here $A$ is the linear operator defined as
$$
(Am)^{(p)}(z_1,...,z_p)=-\frac{1-\delta_{1,p}}{z_1}m^{(p-1)}(z_2,...,z_p)
-\frac{v^2}{z_1}m^{(p+1)}(z_1,z_1,...,z_p)
$$
in the Banach space $B$ of bounded sequences ${\{m^{(p)}\}}_{p=1}^{\infty}$,
$\| m \|=\sup_{p\geq 1} \, v^p |m^{(p)}|$. $f_N^{(p)}$ for every $p$ is a
sum of moments whose form is different from (\ref{2}).
If $|\mbox{Im} \; z_i| > 2v$, then $\|A\| \leq 2v/|\mbox{Im} \; z_i|<1$,
the operator $1-A$ is invertible and (\ref{8}) has the unique solution.
Since according to (\ref{7}) the moments (\ref{2}) are bounded above
by $\prod_{i=1}^p \, |\mbox{Im} \; z_i|^{-1}$, this solution coincides with
$\{m^{(p)}(z_1, \ldots ,z_p)\}_{p=1}^\infty$ under the above condition
$|\mbox{Im} \,z_i|>2v$.
Furthermore, to the leading order $f^{(p)}_N=-\delta_{p,1}/z_1$
and therefore (cf (\ref{3}))
\begin{eqnarray}
m_N^{(p)}(z_1,...,z_p)=\prod^p_{i=1} \, r(z_i) + O(1/N),
\label{9}
\end{eqnarray}
where $r(z)=(-z+\sqrt{z^2-4v^2})/2v^2$ is the Stieltjes transform
of the semicircle law, i.e. $\pi^{-1}\mbox{Im} \; r(E+\mbox{i}0)=
{ }_+ \sqrt{4v^2-E^2}/2\pi v^2$, where ${ }_+ \sqrt{t}=
\sqrt{\max (t,0)}$.
Next orders of $f^{(p)}_N$ in the Gaussian case are given by
expressions
$$
\frac{v^2}{z_1N} \o{ \la G_N(z_1)^2 \ra \prod_{j=2}^p \, \la G_N(z_j) \ra},
\;
\frac{2v^2}{z_1N^2} \sum_{j=2}^p \,
\o{
\la G_N(z_1)G_N(z_j)^2 \ra \prod_{k=2,k\ne j}^p \la G_N(z_k) \ra
},
$$
etc. We see that moments of the form different from (\ref{2})
do appear in these expressions. These new moments of can be
found by an argument analogous to that
for $m_N^{(p)}$, i.e. by deriving equations similar to (\ref{8}).
In general case there are additional terms in each order of $1/N$.
They can be handled analogously (see e.g. the first term
in the r.h.s. of Eq. (\ref{10}) below).
Therefore, solving (\ref{8}) and these "higher order" equations
step by step in each order of $1/N$ we obtain corrections to
$m_N^{(p)}$ for any $p$.
\bigskip
4. For the Gaussian entries $1/N$ corrections were
studied in the physical paper \cite{Verbaarschot_et_al1984}
basing on the formal perturbation theory with respect to $H_{k,l}$
and the diagrammatic technique. This approach is an adaptation
of respective technique developed in \cite{OpermannWegner1979} in
order to construct the $1/N$ expansion of the Green function moments
of the random operator describing a disordered system on $Z^d$ with
$N$ orbitals at each site. It is not an easy problem to extend this
technique (especially in its rigorous version)
to the non-Gaussian case. In comparison with that our approach
is much less sensitive to the type of probability distribution
of $H_{k,l}$. Moreover, the complicated and cumbersome combinatorial
problem of rearranging of diagrams does not appear. In particular,
the "dressing" procedure replacing "bare" Green function $-1/z$ by
$\lim_{N \to \infty} \; \la G_N(z) \ra$ is automatic in our approach.
This is especially evident in the computation of the two-point
correlation function (\ref{4}). Here we can simplify our general procedure
because in this case it is sufficient to iterate only a few first
relations of the infinite system (\ref{8}). Namely, if as before,
$|\mbox{Im} \; z_i| > 2v$, then
\begin{eqnarray}
\left( z_1 - 2v^2 \o { \la G_N(z_1) \ra } \right) C_N(z_1,z_2) & = &
\frac{2\sigma}{N^2} \sum_{k,l=1}^N \;
\o {
G_{k,k}(z_1)G_{k,k}(z_2)G_{l,l}(z_1)G_{l,l}(z_2)^2
} + \nonumber \\
& & \nonumber \\
& &
\frac{2v^2}{N^2} \o {\la G_N(z_1)G_N(z_2)^2 \ra } + O\left( \frac{1}{N^{5/2}} \right)
\label{10}
\end{eqnarray}
\noindent Due to the factor $1/N^2$ in front of the first and the second
terms of the r.h.s. of this relation we can replace respective averages
in these terms by their limiting (zero order) values given by the first term in the
r.h.s. of (8). We obtain after some algebra
\begin{eqnarray}
C_N(z_1,z_2)=N^{-2}F(z_1,z_2)+O(N^{-5/2}),
\label{11}
\end{eqnarray}
\begin{eqnarray}
F(z_1,z_2) & = &
\frac{1}{N^2}
\left(
\frac {2v^2(r_1-r_2)^2}{\beta (z_1-z_2)^2(1-v^2r_1^2)(1-v^2r_2^2)}+
\frac {2\sigma r_1^3r_2^3}{(1-v^2r_1^2)(1-v^2r_2^2)}
\right)+ \nonumber \\
& & \nonumber \\
& &
O\left( \frac{1}{N^{5/2}} \right),
\label{12}
\end{eqnarray}
\noindent where $\beta =1$.
\bigskip
5. Notice that by an analogous argument we can also find $1/N$ corrections
to the $N=\infty$ limit of the Green functions of an ensemble of the Hermitean
random matrices. For instance, let
\begin{eqnarray}
H_{k,l}=N^{-1/2}( X_{k,l} + \mbox{i} Y_{k,l} ) \qquad k,l=1\ldots N,
\label{13}
\end{eqnarray}
\noindent where $X_{k,l}=X_{l,k}$ and $Y_{k,l}=-Y_{l,k}$ are independent
for $k\leq l$ random variables such that i) $\o {X_{k,l}}=\o {Y_{k,l}}=0$;
ii) $\o{X_{k,l}^2} = (1+\delta_{k,l})v^2/2, \;
\o{Y_{k,l}^2} = (1 - \delta_{k,l}) v^2/2$;
iii) $\o {X_{k,l}^4} - 3{(\o {X_{k,l}^2})}^2 = \sigma_X, \;
\o {Y_{k,l}^4}-3{(\o {Y_{k,l}^2})}^2 = \sigma_Y, \;
\sigma_X + \sigma_Y = \sigma,\, \mbox{if} \, k2v$ where they were rigorously proved.
Nevertheless, since the function $F(z_1,z_2)$ given by (\ref{12}) can
obviously be continued up to the real axis with respect to the both variables $z_1$
and $z_2$ we can apply to this expression the operations $I_{E_1}$ and
$I_{E_2}$, $E_1 \ne E_2$. It means that we are going to compute
the leading term of the DOS correlation function by performing
first the limit $N \to \infty$ and then the limits $\epsilon_1, \epsilon_2
\downarrow 0$. This order of limiting transitions is inverse with respect
to that prescribed by the definition of this correlation function.
The resulting expression for the correlation function is
\begin{eqnarray}
S_N(E_1,E_2) & = & -\frac{1}{\beta \pi^2 [N(E_1-E_2)]^2}
\frac {4v^2 - E_1E_2}{\sqrt{4v^2-E_1^2}\sqrt{4v^2-E_2^2}} \nonumber \\
& & \nonumber \\
& &
-\frac {2\sigma}{N^2 \pi^2v^6}\frac{(2v^2-E_1^2)(2v^2-E_2^2)}
{\sqrt{4v^2-E_1^2}\sqrt{4v^2-E_2^2}}.
\label{14}
\end{eqnarray}
\noindent For the Gaussian orthogonal and unitary ensembles (GOE and GUE)
$\sigma =0$ and we recover result
\begin{eqnarray}
S_N(E_1,E_2)=-\frac{1}{\beta \pi^2 [N(E_1-E_2)]^2}
\frac {4v^2 - E_1E_2}{\sqrt{4v^2-E_1^2}\sqrt{4v^2-E_2^2}}.
\label{15}
\end{eqnarray}
\noindent obtained in \cite{French_et_al1978,Pandey1981}. We see
that in a general non-Gaussian case the respective expression
depends not only on the second moment of random entries but also on the
fourth moment via $\sigma$.
Let us consider now the so-called scaling limit of $S_N(E_1,E_2)$,
when $E_1,E_2 \to E, \; N(E_1-E_2) \to s$ \cite{Mehta1991}.
We obtain remarkably simple expression: $\lim_{N(E_1-E_2) \to s}
\; S_N(E_1,E_2) = -1/(\beta \pi^2 s^2)$.
According to Wigner and Dyson (see e.~g. \cite{Mehta1991})
the exact large-$s$ asymptotics
the DOS correlation functions of the Gaussian ensembles are:
$-1/(\pi^2 s^2)$ (GOE) and
$-\sin^2\pi \rho(E) s/(\pi^2 s^2)$ (GUE). Comparing these expressions
with our we see that our procedure of computing of the correlation
function yields for the general case the expression coinciding with the
large-$s$ asymptotics of the Gaussian correlation function smoothed over
energy intervals $\Delta s \gg \rho^{-1}(E)$. This can be regarded as a
support of the universality conjecture of random matrix theory
\cite{Mehta1991}.
Let us mention three more supports of this conjecture.
The first one \cite{MirlinFyodorov1991} concerns the so-called sparse
(or diluted) random matrices whose entries are independently distributed
random variables such that $\mbox{Pr}\, \{H_{k,l}=0\} = p/N$.
The authors used the Grassman integral technique and found the Wigner-Dyson
universal form of the DOS correlator if $p$ is large enough.
The second was obtained
recently \cite{BrezinZee1993} for the completely different ensemble, known
as the unitary invariant ensemble whose probability density is
$Z^{-1}\exp{[-NV(H)]}$, where $V(t)$ is an even polynomial.
Basing on an approach known as the orthogonal polynomial
technique, the authors established a number of interesting
results concerning the eigenvalue statistics of this ensemble,
in particular the relation (\ref{15}) for $\beta =2$.
The third was obtained by present authors for the ensemble
$H = \sum_{\mu =1}^p \, \tau_{\mu}(\cdot,\xi^{\mu})\xi^{\mu}$, where
$\tau_{\mu}$ and $\xi^{\mu}=\{\xi^{\mu}_1, \ldots , \xi^{\mu}_n\}$
are independent identically distributed random variables. For this ensemble
which was introduced in \cite{MarchenkoPastur1967} we obtained the
analogue of (\ref{11}) and (\ref{12}) and showed that its scaling limit is the
same as above. These results will be published elsewhere.
\bigskip
\medskip
\noindent{\large \bf Acknowledgements}
\medskip
\noindent This work was supported in part by the International
Science Foundation under Grant No U2S000 and the State
Committee for Science and Technology of Ukraine under Grant No 3/1/132.
\newpage
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\end{sf}
\end{document}