On Universality of Bulk Local Regime of the Deformed Gaussian Unitary Ensemble

# On Universality of Bulk Local Regime of the Deformed Gaussian Unitary Ensemble

T. Shcherbina
Institute for Low Temperature Physics, Kharkov, Ukraine.
E-mail: t shcherbina@rambler.ru
###### Abstract

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 .

## 1 Introduction.

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

 Nn(△)=♯{λ(n)j∈△,j=¯¯¯¯¯¯¯¯1,n}/n,Nn(R)=1, (1.1)

where is an arbitrary interval of the real axis. For many known random matrices the expectation is absolutely continuous, i.e.,

 ¯¯¯¯¯Nn(△)=∫△ρn(λ)dλ. (1.2)

The non-negative function in (1.2) is called the density of states.

Define also the -point correlation function by the equality:

 E⎧⎨⎩∑j1≠...≠jmφm(λj1,…,λjm)⎫⎬⎭=∫φm(λ1,…,λm)R(n)m(λ1,…,λm)dλ1,…,dλm, (1.3)

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

 N(R)=1,N(△)=∫△ρ(λ)dλ. (1.4)

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

 bulkN={λ|∃(a,b)⊂suppN:λ∈(a,b),ρn(μ)⇉ρ(μ)on(a,b),ρ(λ)≠0}. (1.5)

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

 limn→∞1(nρn(λ0))mR(n)m(λ0+x1ρn(λ0)n,…,λ0+xmρn(λ0)n)=det{S(xi−xj)}mi,j=1, (1.6)

where

 S(xi−xj)=sinπ(xi−xj)π(xi−xj), (1.7)

and , , and are defined in (1.3), (1.2) and (1.4);

(ii) if

 En(△)=P{λ(n)i∉△,i=¯¯¯¯¯¯¯¯1,n}, (1.8)

is the gap probability, then

 limn→∞En([λ0+aρn(λ0)n,λ0+bρn(λ0)n])=det{1−Sa,b}, (1.9)

where the operator is defined on by the formula

 Sa,bf(x)=b∫aS(x−y)f(y)dy,

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)

 Hn=Mn+H(0)n, (1.10)

where is a Hermitian matrix with eigenvalues and is the GUE matrix, defined as

 Mn=n−1/2Wn, (1.11)

where is a Hermitian matrix whose elements and are independent Gaussian random variables such that

 E{Rwjk}=E{Iwjk}=0,E{R2wjk}=E{I2wjk}=12(j≠k),E{w2jj}=1. (1.12)

Let

 N(0)n(△)=♯{h(n)j∈△,j=¯¯¯¯¯¯¯¯1,n}/n.

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 [4] 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

 g(z)=g(0)(z+g(z)).

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 [5]. 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.

This important result was used in [9] to prove universality of the local bulk regime of matrices (1.10), where is a hermitian random matrix with independent (modulo symmetry) entries:

 Wn = {wjk}nj,k,wjk=¯¯¯¯¯¯¯¯wkj (1.13) E{wjk} = E{w2jk}=0,E{|wjk|2}=1,supj,kE{|wjk|p}<∞.

It was proved in [9] that if , then (1.6) is valid, and if , then (1.9) is valid.

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.

###### Theorem 1.

Let be a nonrandom measure that converges weakly to a measure with a bounded support. Then for any the universality properties (1.6 ) and (1.9) hold.

###### Theorem 2.

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

 limn→∞P{|N(0)(Δ)−N(0)n(Δ)|>ε}=0.

Then for any the universality properties (1.6 ) and (1.9) hold.

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], [9] 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 [13]) and widely used in physics literature (see e.g. book [14]). 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.

It is well-known (see for example [5]) that the correlation functions (1.3) for the GUE can be written in the determinant form

 R(n)m(λ1,…,λm)=det{Kn(λi,λj)}, (2.1)

with

 Kn(λi,λj)=n−1∑k=0ϕk(λi)ϕk(λj),ϕk(x)=n1/4hk(√nx)e−nx2/4,

where are orthonormal Hermite polynomials. We want to find analogs of these formulas in the case of random matrices (1.10).

###### Proposition 1.

Let be the random matrix defined in (1.10) and be the correlation function (1.3). Then (2.1) is valid with

 Kn(λ,μ)=−n∫Ldt2π∮Cdv2πexp{−n2(v2−2vλ−t2+2μt))}v−tn∏j=1⎛⎜⎝t−h(n)jv−h(n)j⎞⎟⎠, (2.2)

where is a line parallel to the imaginary axis and lying to the left of all , and the closed contour has all inside and does not intersect .

Representation (2.2) was first obtained in physical papers [6, 7]. We obtain this representation by use the Grassmann integration.

###### Proof.

Following [12], where the GUE was studied, denote

 F(z1,z2,…,zm)=E{m∏k=1Tr1Hn−zk}, (2.3)

where are distinct complex numbers, . It is technically easier to study the ratio of the determinants instead of . Denote

 D(z1,…,zm;x1,…,xm)=det(Hn−z1−x1)…det(Hn−zm−xm)det(Hn−z1)…det(Hn−zm). (2.4)

Since

 −ddxdet(Hn−z−x)det(Hn−z)∣∣∣x=0=Tr(Hn−z)−1,

then

 F(z1,z2,…,zm)=∂m∂x1…∂xmE{D(z1,…,zm;x1,…,xm)}∣∣∣x1=⋯=xm=0. (2.5)

Here and below the symbol denotes the expectation with respect to the measure generated by (see (1.12)).

By using formulas (5.5) and (5.6), we obtain:

 D(z1,…,zm;x1,…,xm)=∫exp⎧⎨⎩−im∑α=1n∑j,k=1(1√nwj,k+δj,k(h(n)j−zα))¯¯¯¯ψj,αψk,α⎫⎬⎭×exp{−im∑α=1n∑j,k=1(1√nwj,k+δj,k(h(n)j−zα−xα))¯¯¯ϕj,αϕk,α}n∏j=1dΦj,

where are the Grassmann variables ( variables for each determinant in the numerator), are complex ones ( variables for each determinant in the denominator), and

 dΦj=1πmm∏α=1(d¯¯¯¯ψj,αdψj,αdRϕj,αdIϕj,α).

Collecting separately the terms with and we get

 ∫exp{im∑α=1n∑j=1(zα−h(n)j)¯¯¯¯ψj,αψj,α+im∑α=1n∑j=1(zα+xα−h(n)j)¯¯¯ϕj,αϕj,α}×exp⎧⎨⎩−i√n∑j

Denote by the first exponential. Integrating with respect to the measure generated by , we obtain after some calculations

 E{D(z1,…,zm;x1,…,xm)}=∫exp{f}⋅exp⎧⎨⎩−12nn∑j,k=1(Φ+jΦk)(Φ+kΦj)⎫⎬⎭n∏j=1dΦj. (2.7)

We will use below the following standart

###### Lemma 1 (Hubbard-Stratonovitch transformation).

We have in the above notations:

 (2.8)

where

 Q=(aσσ+ib),

are Hermitian ordinary matrices, is a matrix consisting of Grassmann variables ( is its Hermitian conjugate), and

 dQ=1πm2m∏j=1daj,jdbj,j∏j
###### Proof.

Define

 s(ψ)α,β=n∑j=1¯¯¯¯ψj,αψj,β,s(ϕ)α,β=n∑j=1¯¯¯ϕj,αϕj,β,s(ψ,ϕ)α,β=n∑j=1¯¯¯¯ψj,αϕj,β,s(ϕ,ψ)α,β=n∑j=1ψj,α¯¯¯ϕj,β.

Write the sum at the exponent as:

 −12nn∑j,k=1(Φ+jΦk)(Φ+kΦj)=−12nm∑α,β=1s(ψ,ϕ)α,β⋅s(ϕ,ψ)α,β−12nm∑α,β=1s(ψ,ϕ)β,α⋅s(ϕ,ψ)β,α−12nm∑α,β=1s(ψ)α,β⋅s(ψ)β,α−12nm∑α,β=1s(ϕ)α,β⋅s(ϕ)β,α (2.9)

Now, use (5.5) to obtain:

 ∫∏α<βdIbα,βdRbα,βπ∏αdbα,απexp⎧⎨⎩m∑α,β=1bαβs(ψ)α,β−n∑α<β¯¯bα,βbα,β−n2m∑α=1b2α,α⎫⎬⎭=∫∏α<βdIbα,βdRbα,βπexp⎧⎨⎩m∑α<βRbαβ(s(ψ)α,β+s(ψ)β,α)⎫⎬⎭×exp{im∑α<βIbαβ(s(ψ)α,β−s(ψ)β,α)−n∑α<β¯¯bα,βbα,β−n2m∑α=1b2α,α}=(√2n)m(√1n)m(m−1)exp⎧⎨⎩12nm∑α,β=1s(ψ)α,βs(ψ)β,α⎫⎬⎭

Similar argument yields the formulas

 ∫∏α<βdIaα,βdRaα,βπ∏αdaα,απexp⎧⎨⎩−im∑α,β=1aαβs(ϕ)α,β−n∑α<β¯¯¯aα,βaα,β−n2m∑α=1a2α,α⎫⎬⎭=(√2n)m(√1n)m(m−1)exp⎧⎨⎩−12nm∑α,β=1s(ϕ)α,βs(ϕ)β,α⎫⎬⎭

and

 ∫m∏α,β=1dηα,βdηα,βexp⎧⎨⎩m∑α,β=1ηαβs(ψ,ϕ)α,β+m∑α,β=1¯¯¯ηαβs(ϕ,ψ)α,β−nm∑α,β=1¯¯¯ηα,βηα,β⎫⎬⎭=m∏α,β=1∫dηα,βdηα,β(1+ηα,βs(ψ,ϕ)α,β+¯¯¯ηα,βs(ϕ,ψ)α,β−n¯¯¯ηα,βηα,β+¯¯¯ηα,βηα,βs(ψ,ϕ)α,βs(ϕ,ψ)α,β)=nm2m∏α,β=1(1−1ns(ψ,ϕ)α,βs(ϕ,ψ)α,β)=nm2exp{−1ns(ψ,ϕ)α,βs(ϕ,ψ)α,β},

where we used (5.6) to obtain the last formula. Collecting together three above formulas, we present the l.h.s. of (2.8) as

 12m∫m∏α,β=1dηα,βdηα,β∏α<βdIaα,βdRaα,βπ∏αdaα,απ∏α<βdIbα,βdRbα,βπ∏αdbα,απ×exp{m∑α,β=1aαβs(ψ)α,β+im∑α,β=1bαβs(ϕ)α,β+m∑α,β=1ηαβs(ψ,ϕ)α,β+m∑α,β=1¯¯¯ηαβs(ϕ,ψ)α,β}×exp⎧⎨⎩−n2m∑α=1(a2αα+b2αα)−n∑α<β(¯¯¯aαβaαβ+¯¯bαβbαβ)−nm∑α,β=1¯¯¯ηαβηαβ⎫⎬⎭=12m∫exp{−n2strQ2}n∏j=1exp{−iΦ+jQΦj}dQ,

where the matrix is described in the lemma. ∎

The above allows us to rewrite the integral in the r.h.s. of (2.7) as:

 12m∫exp{f}⋅n∏j=1dΦj⋅exp{−n2strQ2}n∏j=1exp{−iΦ+jQΦj}dQ. (2.10)

Setting

 Λ=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝z10…000…00z2…000…0……………………0…0zn0…0000…0z1+x10…00…000z2+x2…0……………………0…000…0zn+xn⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,

and using the explicit form of , we obtain from (2.10)

 12m∫n∏j=1dΦj⋅exp{−n2strQ2}n∏j=1exp{−iΦ+j(Q−Λ+hjI)Φj}dQ. (2.11)

Recall now that . Hence, , where is a matrix, whose entries are the real parts of the entries of .

We integrate (2.11) with respect to and by using (5.7), as a result the integral (2.7) is equal to

 12m∫exp{−n2strQ2}n∏j=1% sdet(Q−Λ+hjI)−1dQ=12m∫exp{−n2strQ2}n∏j=1% sdet(Q−Λ1+ε⋅I+hjI)−1dQ=12m∫exp{−n2str(Q+Λ1)2}n∏j=1sdet(Q+ε⋅I+hjI)−1dQ (2.12)

Write , where is a unitary super-matrix and the matrix is

 S=(S100S2),

where

 S1=⎛⎜ ⎜ ⎜⎝s10…000s20…0……………00…0sn⎞⎟ ⎟ ⎟⎠,S2=⎛⎜ ⎜ ⎜⎝it10…000it20…0……………00…0itn⎞⎟ ⎟ ⎟⎠.

Use the super-analog (5.14) of the Itzykson-Zuber formula for the integration over the unitary group (see [12]). This yields

 E{D(z1,…,zm;x1,…,xm)}=1−χ(x1,…,xm)+(−2π)−mnmBm(Λ)∫exp{−n2str(S+Λ1)2}n∏j=1m∏α=1⎛⎜⎝itα+iε+h(n)jsα+iε+h(n)j⎞⎟⎠⋅Bm(S)m∏α=1dtαdsα, (2.13)

where is the Cauchy determinant (5.13).

Using the formula for the Cauchy determinant, we obtain that

 Bm(Λ)−1=m∏α=1xα∏α>β(zα−zβ)(zα+xα−zβ−xβ)(zα−zβ−xβ)(zβ−zα−xα).

Substituting this to (2.13), differentiating (2.13) with respect to every and putting then , we have

 ∂m∂x1…∂xmE{D(z1,…,zm;x1,…,xm)}∣∣∣x1=⋯=xm=0=nm(−2π)m∫exp{−n2str(S+˜Λ)2}n∏j=1m∏α=1⎛⎜⎝it+α+h(n)js+α+h(n)j⎞⎟⎠⋅Bm(S)m∏α=1dtαdsα, (2.14)

where , , .

We can write the determinant (5.13) as

 Bm(S)=∑ω(−1)σ(ω)m∏α=11sα−itω(α),

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

 F(z1,…,zm)=det{ˆKn(zα,zβ)}, (2.15)

where

 ˆKn(zα,zβ)=−n2π∫exp{−n2((sα+Rzα)2−(itβ+Rzβ)2)}n∏j=1⎛⎜⎝it+β+h(n)js+α+h(n)j⎞⎟⎠dtβdsαsα−itβ. (2.16)

Denote

 ˜Kn(λ,μ)==−n2π2∫dtdss−itexp{−n2((s+λ)2−(it+μ)2)}limε→0In∏j=1it++h(n)js++h(n)j=−n2π2∫dtdss−itexp{−n2((s+λ)2−(it+μ)2)}n∏j=1(it+h(n)j)limε→0In∏j=11s++h(n)j. (2.17)

Changing variables to , , we obtain

 ˜Kn(λ,μ)=−in2π2