On Universality of Bulk Local Regime of the Deformed Gaussian Unitary Ensemble
We consider the deformed Gaussian Ensemble in which is a diagonal Hermitian matrix and is the Gaussian Unitary Ensemble (GUE) random matrix. Assuming that the Normalized Counting Measure of (both non-random and random) converges weakly to a measure with a bounded support we prove universality of the local eigenvalue statistics in the bulk of the limiting spectrum of .
Universality is an important topic of the random matrix theory. It deals with statistical properties of eigenvalues of random matrices on intervals whose length tends to zero as . According to the universality hypothesis these properties do not depend to large extent on the ensemble. The hypothesis was formulated in the early 60s and since then was proved in certain cases. There are some results only for special cases. Best of all universality is studied in the case of ensembles with a unitary invariant probability distribution (known also as unitary matrix models) ([1, 2, 3]).
To formulate the universality hypothesis we need some notations and definitions. Denote by the eigenvalues of the random matrix. Define the normalized eigenvalue counting measure (NCM) of the matrix as
where is an arbitrary interval of the real axis. For many known random matrices the expectation is absolutely continuous, i.e.,
The non-negative function in (1.2) is called the density of states.
Define also the -point correlation function by the equality:
where is bounded, continuous and symmetric in its arguments and the summation is over all -tuples of distinct integers . Here and below integrals without limits denote the integration over the whole real axis.
The global regime of the random matrix theory, centered around weak convergence of the normalized counting measure of eigenvalues, is well-studied for many ensembles. It is shown that converges weakly to a non-random limiting measure known as the integrated density of states (IDS). The IDS is normalized to unity and is absolutely continuous
The non-negative function in (1.4) is called the limiting density of states of the ensemble.
We will call the spectrum the support of and define the bulk of the spectrum as
Then the universality hypothesis on the bulk of the spectrum says that for we have:
(i) for any fixed uniformly in varying in any compact set in
is the gap probability, then
where the operator is defined on by the formula
and is defined in (1.7).
In this paper we study universality of the local bulk regime of random matrices of the deformed Gaussian Unitary Ensemble (GUE)
where is a Hermitian matrix with eigenvalues and is the GUE matrix, defined as
where is a Hermitian matrix whose elements and are independent Gaussian random variables such that
be the Normalized Counting Measure of eigenvalues of .
Note also that since the probability law of is unitary invariant we can assume without loss of generality that is diagonal.
The global regime for the ensemble (1.10)-(1.12) is well enough studied. In particular, it was shown in  that if converges weakly with probability 1 to a non-random measure as , then also converges weakly with probability 1 to a non-random measure . Moreover the Stieltjes transforms of and of satisfy the equation
It follows from the definition (1.1) and the above result that any -independent interval of spectral axis such that contains eigenvalues. Thus, to deal with a finite number of eigenvalues as , in particular, with the gap probability, one has to consider spectral intervals, whose length tends to zero as . In particular, in the local bulk regime we are about intervals of the length .
Random matrix theory posseses a powerful techniques of analysis of the local regime, based on the so called determinant formulas for the correlation functions . For the GUE, more general for the hermitian matrix models, the determinant formulas follow from the possibility to write the joint probability density of its eigenvalues as the square of the determinant, formed by certain orthogonal polynomials and then as the determinant formed by reproducing kernel of the polynomials, that.are also heavily used in the subsequent asymptotic analysis [1, 2, 3]. Unfortunately, the orthogonal polynomials have not appeared so far in the study of the deformed Gaussian Unitary Ensemble. However, it was shown in physical papers [6, 7, 8] that correlation functions of the deformed Gaussian Unitary Ensemble can be written in the determinant form, although the corresponding kernel is not, in general, a reproducing kernel of a system of orthogonal polynomials. This was done by using as a crucial step the Harish-Chandra/Itzykson-Zuber formula for certain integrals over the unitary group.
Later in the series of the papers [10, 11] the special case of (1.10 ) was studied, where has two eigenvalues of equal multiplicity. In this case universality in the bulk and at the edge of the spectrum were proved.
In this paper we consider random matrices (1.10) for a rather general class of both random and nonrandom. The main results are the following theorems.
Let the eigenvalues of in (1.10) be a collection of random variables independent of and such that (the symbol denotes the expectation with respect to the measure generated by ). Assume that there exists a non-random measure of a bounded support such that for any finite interval and for any
The paper is organized as follows. In Section we give a new proof of determinant formulas for correlation functions (1.3) by the method which is different from those of [6, 7],  and [10, 11]. Namely we use the representation of the resolvent of the random matrix via the integral with respect to the Grassmann variables. The integral was introduced by Berezin (see ) and widely used in physics literature (see e.g. book ). For the reader convenience we give in Appendix a brief account of the Grassmann integral techniques that we will use in the paper. Theorem 1 will be proved in Sections . Section deals with the proof of Theorem 2.
2 The determinant formulas.
where are orthonormal Hermite polynomials. We want to find analogs of these formulas in the case of random matrices (1.10).
Following , where the GUE was studied, denote
where are distinct complex numbers, . It is technically easier to study the ratio of the determinants instead of . Denote
Here and below the symbol denotes the expectation with respect to the measure generated by (see (1.12)).
where are the Grassmann variables ( variables for each determinant in the numerator), are complex ones ( variables for each determinant in the denominator), and
Collecting separately the terms with and we get
Denote by the first exponential. Integrating with respect to the measure generated by , we obtain after some calculations
We will use below the following standart
Lemma 1 (Hubbard-Stratonovitch transformation).
We have in the above notations:
are Hermitian ordinary matrices, is a matrix consisting of Grassmann variables ( is its Hermitian conjugate), and
The above allows us to rewrite the integral in the r.h.s. of (2.7) as:
and using the explicit form of , we obtain from (2.10)
Recall now that . Hence, , where is a matrix, whose entries are the real parts of the entries of .
Write , where is a unitary super-matrix and the matrix is
where is the Cauchy determinant (5.13).
Using the formula for the Cauchy determinant, we obtain that
where , , .
We can write the determinant (5.13) as
where the sum is over all permutations of the indices , and is the parity of a permutation. The rest of the integrand factorizes in a -fold product. Hence, recalling the definition of in (2.5), we obtain finally
Changing variables to , , we obtain