eigenvalue correlations for generalized gaussian ensembles

Physica A 288 (2000) 119–129www.elsevier.com/locate/physa

Eigenvalue correlations for generalizedGaussian ensembles

Pragya ShuklaDepartment of Physics, Indian Institute of Technology, Kharagpur, West Bengal, India

Abstract

We study the statistical properties of the eigenvalues of a generalized Gaussian ensemble ofHermitian matrices described by a set of the variances of the matrix elements and non-invariantunder an unitary transformation. We �nd that the eigenvalue distribution in this case can bemapped on to the corresponding distribution in Dyson’s Brownian ensembles, with a functionof variances in the former playing the role of “time” in the latter. The pre-existing informationabout the spectral correlations of Brownian ensembles can therefore be used to obtain the samefor generalized Gaussian ensembles. c© 2000 Elsevier Science B.V. All rights reserved.

PACS: 05.45+b; 03.65.sq; 05.40+j

1. Introduction

The ensemble of Gaussian random matrices can generally be described as that oflarge N × N matrices with a Gaussian distribution for their matrix elements. Suchstructure quite often appear in various physical contexts related to complex systemse.g. Hamiltonian and, therefore, need to be analyzed in great detail [1]. However thenature and degree of complexity of the physical system may give rise to various typesof structures. For example, for the delocalized nature of quantum dynamics in a certainspace, the Hamiltonian matrix constructed by using the set of eigenfunctions associ-ated with this space as basis, turns out to be a full random matrix. Here almost allthe matrix elements are of the same order. On the other hand, the localized quantumdynamics can give rise to many matrix elements with diminishing strength. For lo-calization in one dimension, the non-zero elements are e�ectively con�ned within aband of width b¿ 1 around the main diagonal; such matrices are termed as randombanded matrices (RBM). The random banded matrix ensembles, with zero mean valueof all the matrix elements and the variance 〈|Hij|2〉˙ a(|i − j|), can serve as a good

0378-4371/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(00)00418 -0

120 P. Shukla / Physica A 288 (2000) 119–129

model for the spectral statistics of the tight-binding Hamiltonian of a quantum particlein a 1-D system with long-range random hoppings [2] and also for quasi 1-D disorderedwires [3]. Here the function a(r), re ecting the nature of localization of the associatedeigenfunctions [4–6], decays with r either as a power law or exponentially (or faster)at r/1. Another example is that of Wigner random banded matrices [7,8] where meanvalue of the diagonal elements increases linearly along the main diagonal (〈Hnn〉=�n).These matrices have been shown to be a good model, for example, for the tight-bindingHamiltonian of a quantum particle in a 1-D disordered system subjected to a constantelectric �eld as well as for heavy atoms and nuclei [9]. The ensemble of bandedmatrices with diagonal elements uctuating much stronger than the o�-diagonal ones(〈|Hii|2〉=〈|Hij|2〉˙ b/1), also known as RBM with preferential basis, are quite usefulfor the physical understanding of the two interacting particles propagating in a quenchedrandom potential [10]. The random banded matrix ensembles with correlated diagonalscan well-model the level statistics in a single Landau level of quantum Hall systemswith disorder [11]. The localization in three-dimension, e.g. in the case of a metal–insulator transition in the disordered systems, gives rise to a structure, richer thanRBM, known as random sparse matrix (RSM), in which each row contains an averagenumber of non-zero matrix elements. Another type of matrices which appear in physicalsystems are the periodic ones which can be described as the matrix M of the rank Nsuch that Mij 6= 0 only if |i − j|6b or if |i − j|¿N − b [12,6]. Here all the non-zeroelements belong to three regions: a band of half-width b along the main diagonal,upper right corner and lower left one. Such a matrix can describe the motion along adiscrete ring of N sites, with non-zero hopping amplitude only for transitions betweensites which lie no more than b sites apart along the circle.Many important physical properties of complex quantum systems can be studied by

analyzing the statistical behavior of the eigenvalues of associated quantum operators.Given the probability distribution of the matrix elements in an ensemble, the eigenvaluedistribution can be obtained by averaging out the information regarding the eigenvec-tors. The latter is easier to be performed if the probability distribution of the matrixelements is invariant under the change of the basis (and therefore, independent of theeigenvector components) [13]. However, for non-invariant distributions, the required in-tegration over eigenvectors is, in general, a formidable task. The techniques suggestedso far for this purpose e.g non-linear sigma model [2,9] depend strongly on the typeof distribution chosen and mostly give approximate results. Our aim in this paper isto suggest a method which leads to results in a generic form valid for most of theGaussian RM ensembles (GGE). This we achieve by mapping the problem to the studyof a particular class of transition ensembles, namely, Dyson’s Brownian ensembles [13].The latter, characterized by a �ctitious ‘time’ t and a �ctitious ‘temperature’ �−1, de-scribe non-stationary states of eigenvalue distribution and yield the stationary ensemblesfor large t-values. For real systems, t and � represent the strength and the symmetryclass of the random perturbation, respectively, which a�ects the eigenvalue-statisticsof a quantum operator e.g. Hamiltonian, time-evolution operator etc. As showed byDyson, in such ensembles, the dynamics of eigenvalues, constrained to move on a real

P. Shukla / Physica A 288 (2000) 119–129 121

line in GE, is identical to that of a set of charged particles, moving under the mutualrepulsion of a two-dimensional coulomb potential and executing Brownian motion intime t (also referred as Wigner–Dyson gas). Here the transition to equilibrium, with tas the evolution parameter, is rapid, discontinuous for in�nite dimensions of matrices.However, the transition in the eigenvalue correlations is governed, for small-t and largematrix dimensionality N , by a parameter � which measures locally the mean-squaresymmetry breaking matrix element (o�-diagonal) in units of the average level spacing.As shown in this paper, the “time” t for the generalized Gaussian ensemble case can beidenti�ed with a function of the distribution parameters of the matrix elements, a vari-ation which can lead to a transition from one class of generalized Gaussian ensemblesto another.The paper is organized as follows. We �rst study the evolution of the eigenvalue

distribution of H , taken from a generalized Gaussian ensemble of Hermitian matrices,due to the variation of the variances of its matrix elements, described in Section 2.We obtain a partial di�erential equation which, after certain parametric rede�nitions,turns out to be formally the same as the Fokker–Planck equation governing the evolu-tion of Wigner–Dyson (WD) gas. Section 3 deals with the solution of the F–P equationas well as various correlations, obtained by the same method as in the case of Brow-nian ensembles. In Section 4, we apply our method to derive the results for a few ofthe generalized Gaussian ensembles. We conclude in Section 5 which is followed byan appendix containing the proof of one of the formulas used in this paper.

2. Equation for the evolution of eigenvalue distribution

The distribution of the matrix elements with zero mean in a Gaussian ensembleof Hermitian (real-symmetric) matrices can, in general, be described as �(H; y) =C exp(−∑k6l �klH

2kl) with C as the normalization constant and y as the set of the

coe�cients ykl= �klgkl with gkl=1+ �kl. Let P(�; y |H0) be the probability of �ndingeigenvalues �i of H between �i and �i + d�i at a given y (with H0 as an initialcondition),

P(�; y|H0) =∫ N∏

i=1

�(�i − �i)�(H; y) dH : (1)

As the �-dependence of P in Eq. (1) enters only through �(H) and @�=@�kl=[(2�kl)−1−H 2kl]�= (2�kl)

−1[�+Hkl@�=@Hkl], a derivative of P with respect to �kl can be writtenas follows:

@P@�kl

=P2�kl

+12�kl

∫ N∏i=1

�(�i − �i)Hkl @�@Hkl dH : (2)

The integration by parts of the integral, followed by a use of the equality @∏Ni=1 �(�i−

�i)=@Hkl =−∑Nn=1(@

∏Ni=1 �(�i − �i)=@�n) @�n=@Hkl, then gives 2�kl@P(�; y)=@�kl = Ikl,


where

Ikl =N∑n=1

@@�n

∫ N∏i=1

�(�i − �i) @�n@HklHkl�(H) dH : (3)

Our aim is to �nd a function Y of the coe�cients �kl’s such that the evolution ofP(�; Y ) in terms of Y satis�es a F–P equation similar to that of Dyson’s Brownianmotion model (Wigner–Dyson gas) [13]. For this purpose, we consider the sum 2∑

k6l( − gkl�kl)�kl@P=@�kl which can be rewritten as follows

2∑k6l

( − ykl)ykl @P@ykl = ∑k6l

Ikl −∑k6l

yklIkl (4)

here is an arbitrary parameter and can be chosen, for reasons to become clear lateron, as the minimum value taken by any of the ykl’s.As �n =

∑i; j OinHijOjn = 2

∑i6j OinHijOjn=gij (eigenvalue equation, O being the

orthogonal, eigenvector matrix), one can obtain the rate of change of �n with respectto Hkl: @�n=@Hkl = (2=gkl)OknOln. Using this relation, the �rst term on the right-handside (Eq. (4)) can further be simpli�ed,∑

k6l

Ikl =∑n

@@�n

(�nP) : (5)

The second term can similarly be rewritten as follows:

∑k6l

yklIkl =−N∑n=1

@@�n

∑k6l

∫ N∏i=1

�(�i − �i)OknOln @�(H)@HkldH

=N∑n=1

@@�n

∑k6l

∫@@Hkl

[N∏i=1

�(�i − �i)OknOln]�(H) dH : (6)

Eq. (6) can further be reduced by di�erentiating the terms inside the brackets, givingus two integrals. The use of orthogonality relation of matrix O in the �rst integral soobtained and the relation (Appendix A)∑

k6l

@OnkOnl@Hkl

=∑m 6=n

1�n − �m (7)

in the second integral gives

∑k6l

yklIkl =−∑n

@@�n

@@�n

+∑m 6=n

��m − �n

P : (8)

Using both Eqs. (5) and (8) in Eq. (4) we obtain the desired F–P equation

@P@Y

= ∑n

@@�n

(�nP) +∑n

@@�n

@@�n

+∑m 6=n

��m − �n

P (9)


where �=1 and the left-hand side of the above equation, summing over all ykl’s, hasbeen rewritten as @P=@Y with Y given by the condition that

@@Y

= 2∑k6l

ykl( − ykl) @@ykl

: (10)

By using the unitarity of eigenvectors and following the same steps, it can be provedfor complex Hermitian case too (now � = 2); this case is discussed in detail inRef. [14].The de�nition of Y depends on the relative values of the coe�cients �kl’s and can be

obtained as follows. We de�ne M =N (N +1)=2 variables (Y1; : : : ; YM ) as the functionsof all ykl’s such that the condition C is satis�ed. This is indeed possible by usingthe orthogonal (Jacobi) coordinate transformation between variables {Yi}i=1; :::;M and{Xi}i=1; :::;M given by Y ≡ Y1 = (1=M)

∑Mj=1 Xj and Yi =

∑Mj=1 aijXj for i = 2→M

with Xj = 12 lnyj=|yj − | + cj for j = 1; : : : ; N (N + 1)=2 with cj as arbitrary constants

of integration. Here coe�cients aij must satisfy the relation∑M

j=1 aij = �i1 which is anecessary condition for orthogonality but not su�cient to get the right form for @=@Y .With D being the functional derivative of Yi’s with respect to Xj’s, we also need theelements D−1

ij of its inverse to be unity. One way to achieve this is to set all adjunctsof the matrix elements @Yi=@Xj equal. The form of Y , ful�lling condition (10), cantherefore be given as Y = [1=N (N + 1)]

∑k6l lnykl=|ykl − |+ C with C =

∑j cj.

The steady state of Eq. (9), P(�;∞) ≡ P∞=∏i¡j |�i−�j|�e−( =2)

∑k �

2k , is achieved

for Y −Y0→∞ (with Y0 as the value of Y for initial ensemble H0) which correspondsto almost all ykl→ . This indicates that, in the steady state limit, the system tendsto belong to the standard Gaussian ensembles (SGE). Eq. (9) (later referred as thevariance–variation or VV case) is formally the same as the F–P equation governingthe Brownian motion of particles in Wigner–Dyon gas [13] with the transition parameterbeing Y in the former and time in the latter. This is also similar to the F–P equationfor the eigenvalue distribution of a hermitian matrix H =H0 + �V undergoing randomperturbation V , of strength �, taken from a standard Gaussian ensembles (〈V 2ii 〉=2〈V 2ij〉=1) with arbitrary initial condition H0 (later referred as perturbation variation or PVcase [15–17], also known as the Rosenzweig Porter or RP model); here � acts as thetransition parameter.It must be noted here that Eq. (7) and, therefore, evolution equation (9) of P(�; Y )

is no longer valid if the matrix H is in a block-diagonal form. For this case, theevolution of eigenvalues in each block can be considered separately, leading to oneF–P equation similar to Eq. (9) for each block. This can also be seen as follows. Therelation corresponding to Eq. (7) for this case is

∑k6l

@OnkOnl@Hkl

=s∑b=1

∑m 6=n

1�bn − �bm

(11)

where “s” corresponds to the total number of blocks and the superscript b on thequantities refers to the those corresponding to a block b. As all other steps, givenabove, remain una�ected by block-diagonal form of H , the use of this relation in


Eq. (6), instead of Eq. (7), leads to the following equation for the joint distribution ofall the eigenvalues of H (for a given initial condition):

∑b

@P@Y b

=N∑b=1

∑n

@@�bn

b�bn @@�bn +

∑m 6=n

��bm − �bn

+@@�bn

P ; (12)

where b is the minimum value among the coe�cients of the matrix elements in theblock b and Y b is given by the condition

@@Y b

= 2∑

k6l;k; l∈block bykl( − ykl) @

@ykl: (13)

As indicated by Eq. (12), the eigenvalues belonging to di�erent blocks do not repeleach other, are not correlated and undergo an evolution independent of the other block.

3. Calculation of P(�; �) and correlations

In the preceding section, we obtained a F–P equation governing the evolution ofprobability P(�; Y |H0) for an arbitrarily given H0 (the conditional probability). The for-mal similarity of the F–P equation with that of WD gas as well as perturbation-variationcase makes it easier to obtain P(�; Y ) at least for �= 2 as the solution of Eq. (9) forthis �-value is already known [16,17]. For completeness purpose, We give here few ofthe steps, used in the solving perturbation-variation case, for our case; for details referto Refs. [16,17].As can readily be checked, Eq. (9) can also be written as follows (with = 1, for

simpli�cation)@P@Y

=∑n

@@�n

|QN |� @@�nP

|QN |� ; (14)

where |QN |� = |�(�)|�e−(1=2)∑

k�2k with �(�) =

∏i¡j(�i − �j). The transformation

= P=|QN |�=2 allows us to cast Eq. (14) in the suggestive form@@Y

=−H ; (15)

where the ‘Hamiltonian’ H turns out to be the Calogero–Moser Hamiltonian

H =∑i

@2

@�2i− 12

∑i¡j

�(� − 2)(�i − �j)2 +

14

∑i

�2i : (16)

With the parabolic-con�ning potential and under the requirement (to take account ofthe singularity in H) that the solutions vanish as |�i − �j|�=2 when �i and �j are closeto each other, H , in Eq. (15), has well-de�ned (completely symmetric) eigenstates �kand eigenvalues �k . This allows us to express the “state” and therefore P(�; Y |H0)as a sum over eigenvalues and eigenfunctions of H

P(�; Y |H (Y0)) =∣∣∣∣ QN (�)QN (�0)

∣∣∣∣�=2∑

k¿0

exp[− �k(Y − Y0)]�k(�)�∗k (�0) ; (17)


where �0 ≡ (�01; �02; : : : ; �0N ) are the eigenvalues of H (Y0). The joint probabilitydistribution P(�; Y ) can then be obtained by integrating over all initial conditions

P(�; Y ) =∫P(�; Y |�0; Y0)P(�0; Y0) d�0 (18)

which further leads to the correlations Rn(�1; : : : ; �n;Y ) = N !=(N − n)!∫d�n+1 : : :

d�NP(�1; : : : ; �N ;Y ) using standard techniques [16,17]. In fact, a direct integration ofF–P equation (9) leads to the BBGKY hierarchic relations among the unfolded localcorrelators Rn(r1; : : : ; rn;�) = LimN→∞Rn(�1; : : : ; �n;Y )=R1(�1;Y ) : : : R1(�n;Y ) withr =

∫ r R1(�;Y )dY and �= (Y − Y0)=D2 (D1 = R−1; the mean level spacing) [16,17]@Rn@�

=∑j

@2Rn@r2j

− �∑j 6= k

@@rj

(Rn

rj − rk

)− �

∑j

@@rj

∫ ∞

−∞

Rn+1rj − rk : (19)

By using the above equation, it can be shown that the transition for Rn occurs onthe scales determined by Y ≈D2 while for R1, the corresponding scale is given byY ≈ND2. This implies, therefore, during the transition in Rn, the density R1 remainsnearly unchanged; this fact is very helpful in unfolding the correlators Rn. For n = 1and in large N -limit, above equation reduces to the Dyson–Pastur equation [16,17] forthe level density, �(�1; Y ) ≡ N−1R1,

@�@Y

=−� @@�

(∑m

P∫d�′

�m� − �′

)� ; (20)

which results in a semi-circular form for � (thus agreeing well with the � obtained in[18] by super-symmetry technique). For n=2 and small values of r, the integral termin Eq. (19) makes a negligible contribution thus leading to the following approximatedclosed-form equation for R2.

@R2@�

= 2@2R2@r2

− 2� @@rR2r: (21)

For large r, the integral term cannot be neglected and, therefore, Eq. (19) cannot beeasily solved. However, for �=2, an exact solution of Eq. (19) for n=2 with variousinitial condition is obtained in Refs. [16,17].Eqs. (17) and (18) represent the formal solutions of Eq. (9). To proceed further,

one needs to know the eigenvalues and eigenfunctions of H so as to express P in acompact form but these are explicitly known only for � = 2 case [13,16,17]. This isbecause, for � = 2 the interaction term in Eq. (16) drops out and P can explicitly beobtained. For CM model (Eq. (16)), it is given as follows [13].

P(�; Y |H (Y0); � = 2)˙ | �(�)�(�0)

|det[f(�i − �0j;Y − Y0)]i; j=1 ::: N (22)

with f(x − y; t) = exp[− (x − y)=t2].As we already know P(�; Y ) and various correlations for Wigner–Dyson case (for

which Y ≡ t) starting from various initial conditions [16,17], one can obtain these mea-sures for variance–variation case just by substituting Y by its appropriate relationshipwith the variances of the matrix elements of H .


4. Examples

In general, to �nd the correlations of a generalized Gaussian ensemble H (non-blockdiagonal form) with given yij’s ¿ , it is mapped to a Brownian ensemble which ap-pears in a transition where the �nal ensemble, referred here as F , is a member of thestandard gaussian ensembles with P(F) = e− Tr F

2and the initial one, referred here as

O, as that of the diagonal matrices with a Poisson distribution of the eigenvalues. Thischoice, therefore, corresponds to the initial values of �ii’s as �nite and �ij;i 6= j→∞giving Y0 = −[1=N (N + 1)]∑N

i=1 ln|1 − yii[O]−1| and the correlation of H will beequivalent to a Brownian ensemble, corresponding to “time” t ≡ Y in the Poisson→standard gaussian ensembles transition (the perturbation–variation case). However, asmentioned previously, even in the perturbation–variation case, the two-point correlationR2(r;�) is exactly known only when the transition occurs due to a perturbation be-longing to GUE [16,17] and, as discussed above, can be used for the variance–variationcase if the matrix H is a complex Hermitian (with �= (Y − Y0)=D2).

R2(r;�)− R2(r;∞) = 4�∫ ∞

0dx∫ 1

−1dz cos(2�rx) exp[− 8�2�x(1 + x + 2z√x)]

×(√

(1− z2)(1 + 2z√x)1 + x + 2z

√x

);

where R2(r;∞) = 1− sin2(�r)=�2r2 (the GUE limit). As can easily be checked, aboveequation has the correct limiting behavior, that is, R2 = 0 for �→ 0 (the Poisson caseY = Y0) and R2 = R2(r;∞) for �→∞. As obvious from Eq. (23), R2 depends on Yand therefore on the nature of variances which, for �nite �-values (in limit N→∞),results in various types of the level-statistics intermediate between Poisson and GUE.However, if the generalized gaussian ensemble under consideration is in a block-

diagonal form, both the initial as well as �nal ensemble should also be chosen in ablock-diagonal form, with each block independently belonging to a Poisson ensemblefor �→ 0, and, to standard gaussian ensemble for �→∞. The ensemble H , therefore,is mapped to an ensemble of matrices in the block-diagonal form with each blockbelonging to a Brownian ensemble (t ≡ Yb) present in Poisson→GE transition.For further clari�cation, we apply our technique to some of the well-known Gaussian

hermitian ensembles.Case 1: All the o�-diagonal matrix elements with same variance but di�erent from

the diagonal ones (Rosenzweig–Porter model).Let us choose �ij;i 6= j = 2(1 + �) and �ii = 1 with �¿0; thus = 2. This ensemble

H can, therefore, be mapped to a Brownian ensemble, with Y − Y0 =−[(N − 1)=(N +1)] ln|1−1=(1+�)| ≈ 1=2� for �¿ 1, appearing in a Poisson → GOE transition wherethe initial matrix elements distribution is given by P(0)˙ e−(1=2)

∑i O

2ii and the �nal,

stationary state, obtained for large �≈ 1=2�D2-values, is P(F)˙ e−(1=2)Tr F2. Now as

R1≈√N [15,18], the D2≈ 1=N and, therefore, �≈N=2� which implies that H will

have an eigenvalue statistics very di�erent from that of the Poisson or GOE only if


�≈ cN (c a �nite constant). For �¿cN , �→ 0 and for �¡cN , �→∞ for N→∞and, thus, H behaves like a Poisson ensemble in the �rst case and like a GOE inthe second. (Note in Ref. [20], D is taken as D˙ 1=N , which gives �≈N 2=2� andtherefore GOE and Poisson ensemble result for �¡cN 2 and �¿cN 2; respectively; thisresult is in agreement with the one obtained, in Ref. [20], by using NLSM techniqueensemble in large N -limit).Case 2: Matrix elements with exponential decay of variancesFor the exponential decay of variances away from the diagonal i.e., �kl=e|k−l|=b; k6l;

1.b.N . Thus, again = 2 and the �nal ensemble is a SGE with P(H) = e−( =2)Tr H 2

and, therefore, Y =−[1=N (N +1)]∑Ni6j=1 ln|1− y−1ij |+C. Here the initial ensemble

is that of the diagonal matrices with a Poisson distribution of the eigenvalues whichcorresponds to yii(O) = 2 and yij;i 6=j(O)→∞ (this being maximum value of ykl(G))giving Y0 = −[1=N (N + 1)]∑N

i=1 ln|1 − y−1ii (O)| + C. Thus, Y − Y0 = −[1=N (N +1)]∑N

r=1(N − r)ln|1− 2e−r=b| ≈ b=N . As R1≈√N , the transition parameter for in�nite

system (N→∞) turn out to be �=Y=D2≈ b which recon�rms that, in in�nite systems,the transition is governed only by the bandwidth b [21,12].Case 3: Matrix elements with power-law decay of variancesFor this case Hij = Gija(|i − j|) with G a typical member of standard gaussian

ensembles and a(r) = 1 and (b=r)� for r6b and ¿b (b/1); respectively (known asPRBM model with P stands for power). Again as before, the choice of the initialand the �nal ensemble remain the same. Now as yij ≡ yr = (r=b)2� (with r= |i− j|),we get

� = D−2(Y − Y0) =− 1(N + 1)

N∑r=b+1

(N − r)ln(1−

(br

)2�)

≈ N2

∞∑j=1

1j

(bN

)2j� ∫ 1

b=Ndx(1− x)x−2j�

=N2

∞∑j=1

1j

[1

2(1− 2j�)(1− j�)(bN

)2j�

− 1(1− 2j�)

bN+

12(1− j�)

(bN

)2]: (24)

Thus, for large N -values and �¡ 12 , �(˙ N 1−2�) is su�ciently large and the eigen-

value statistics approaches SG limit. At �=12 , the statistics is governed by the parameter

b2=N instead of N only.Case 4: Matrix elements in a block-diagonal form.An example of this case can be that of a real-symmetric matrix with two blocks,

both of equal size N1 (where 2N1 = N ), referred by H 1 and H 2 and labeled by b =1 and 2. Let the matrix elements distribution in the block b be given by P(Hb)=exp[ − (�b=2) Tr(H 2b )]. Now, by choosing the initial blocks H 10 and H 20 distributed as


P(Hb0 ) ˙ exp[ − (�b=2)∑

i H2ii ], we can easily obtain the parameter Y

b − Y b0 corre-sponding to each block

Y b − Y b0 =(N1 − 1)(N1 + 1)

ln�b

( b − �b) : (25)

But as b=�b, the �b=(Y b−Y b0 )=D2→∞ and, therefore, the eigenvalue distributionof each block will correspond to a GOE. As the distribution of eigenvalues in boththe blocks is described by the independent F–P equations for each one, our methodgives the correct result: the spectra of matrix H corresponds to that of the 2-GOEssuperimposed on each other.

5. Conclusion

In this paper, we have suggested a technique to obtain the eigenvalue correlationsof an ensemble of Gaussian–Hermitian matrices with matrix elements distributed witharbitrary variances. By incorporating the in uence of all the variances on the eigen-values in a single parameter Y , we reduce this complicated problem to the study ofDyson’s Brownian ensembles. It is clear, therefore, that all we need to determine is thedetailed information about all the correlations of the Brownian ensembles. Althoughso far, very few exact results are known for these ensembles, nonetheless, it is stillfar more tractable than a separate analysis of each of the various types of generalizedGaussian ensembles. Our technique can also be extended to the situation where anunderstanding of the sensitivity of the eigenvalues of a given hermitian operator to theexternal perturbations of generalized Gaussian ensemble type is desired [19,14].As mentioned in Section 2, the generalized gaussian ensembles appearing in various

physical contexts are far more rich in structure and can have correlated matrix ele-ments with non-zero mean. We believe that our technique can be extended to includesuch cases also and the progress made so far in this direction seems to support ourbelief. Furthermore, certain matrix structures related to some physical contexts mayhave non-Gaussian distributions too e.g. P(H) ˙ exp[ − f(H)] with f as somewell-behaved function of H . It is desirable to see as to how well our technique isapplicable to deal with such cases [14].

Acknowledgements

I am grateful to B.S. Shastry, B.L. Altshuler, V. Kravtsov and N. Kumar for varioususeful suggestions during the course of this study.

Appendix A

Proof of equality.∑

k6l @OnkOnl=@Hkl =∑

m 1=�n − �m.


The use of the eigenvalue equation HO = O�, with O as a orthogonal matrix and� the eigenvalue matrix, leads to the following:∑

j

HijOjn = �nOin : (A.1)

Di�erentiating both sides of the above equation with respect to Hkl, we get∑j

@Ojn@Hkl

Hij +∑j

Ojn@Hij@Hkl

= �n@Oin@Hkl

+@�n@Hkl

Oin : (A.2)

Now, multiplying both the sides by Omi (m 6= n) followed by a summation over alli’s, we get the following∑

j

Omj@Ojn@Hkl

=1

�m − �n∑i; j

Omi@Hij@Hkl

Ojn ; (A.3)

a multiplication of both the sides by Omr followed by a summation over all m’s thengives

@Onr@Hkl

=1gkl

∑m

Omr�n − �m (OmkOnl + OmlOnk) ; (A.4)

a multiplication of both the sides by Onk followed by a summation over all k6l thengives ∑

k6l

Onk@Onl@Hkl

=∑m

1�n − �m

∑k6l

OnkOmlgkl

(OmkOnl + OmlOnk) : (A.5)

Now, by using the orthogonality relation∑

j OjkOjl=�kl, one can obtain the desiredrelation.

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eigenvalue correlations for generalized gaussian ensembles

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