Statistical properties of eigenvectors in non-hermitian Gaussian random matrix ensembles

Transcription

1 JOURNAL OF MATHEMATICAL PHYSICS VOLUME 4, NUMBER 5 MAY 2000 Statistical properties of eigenvectors in non-hermitian Gaussian random matrix ensembles B. Mehlig a) and J. T. Chalker Theoretical Physics, University of Oxford, Keble Road, OX 3NP, United Kingdom Received 23 June 999; accepted for publication 20 September 999 Statistical properties of eigenvectors in non-hermitian random matrix ensembles are discussed, with an emphasis on correlations between left and right eigenvectors. Two approaches are described. One is an exact calculation for Ginibre s ensemble, in which each matrix element is an independent, identically distributed Gaussian complex random variable. The other is a simpler calculation using N as an expansion parameter, where N is the rank of the random matrix: this is applied to Girko s ensemble. Consequences of eigenvector correlations which may be of physical importance in applications are also discussed. It is shown that eigenvalues are much more sensitive to perturbations than in the corresponding Hermitian random matrix ensembles. It is also shown that, in problems with time evolution governed by a non-hermitian random matrix, transients are controlled by eigenvector correlations American Institute of Physics. S I. INTRODUCTION Hermitian random matrices have been very successfully used to model Hamiltonian operators of closed quantum systems. In many cases, this has lead to a quantitative description of features such as spectral fluctuations in classically chaotic quantum systems and in disordered quantum systems in the metallic regime. 2 Within this approach it is also possible to describe the statistical properties of wave functions and matrix elements in such systems. Random matrices are also of great importance in many other areas of physics in which they are not constrained to be Hermitian. 3,4 These include: the dynamics of neural networks, 5 the quantum mechanics of open systems, 6 classical diffusion in random media, 7 and in population biology, 8 and modeling the statistical properties of flux lines in superconductors with columnar disorder. 9 3 Recently, in connection with these problems, spectral properties of non-hermitian random matrices and operators have been studied in great detail see, for instance, Refs. 3,4,7,8,4 8. In the context of fluid dynamics, it is well known 9 2 that systems governed by non- Hermitian evolution operators exhibit striking features. First, such systems are particularly sensitive to perturbations. Second, these systems can exhibit pseudoresonances at which the system reacts strongly to an external perturbation although the excitation frequency is not close to any of the frequencies of the internal modes. Third, non-hermiticity can give rise to interesting transient features in time evolution. Such features cannot be understood solely in terms of the spectrum of the evolution operator. While the eigenvalues of the evolution operator determine the long-time behavior, transients and sensitivity to perturbations, in particular, are determined by the properties of the corresponding eigenvectors. In this paper, we quantify the statistical properties of the eigenvectors of random non- Hermitian matrices and examine to what extent enhanced sensitivity to perturbations and transients in time evolution are present in random systems described by non-hermitian operators, such as Fokker Planck operators 7 or projected Hamiltonians. 6 We report on results for two ensembles of random matrices, namely, Ginibre s ensemble 3 and Girko s ensemble. 4 There are several reaa Present address: Max-Planck-Institute for the physics of complex systems, Nöthnitzer Str. 38, 087 Dresden, Germany /2000/4(5)/3233/24/$ American Institute of Physics

2 3234 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker sons for studying these ensembles. First, characterizing the statistical properties of eigenvectors in these cases is by itself a problem of considerable interest: we show that left and right eigenvectors exhibit striking correlations, which depend strongly on where in the spectrum the corresponding eigenvalues lie. Second, non-hermitian operators such as the Fokker Planck operator governing classical diffusion in a random velocity field 7 may be represented, in a finite system and in an appropriate basis, as random matrices. In general, their matrix elements exhibit certain structures and are much less uniform than the matrices from the ensembles we investigate. Nevertheless, experience of universality in random Hermitian problems gives reason to hope that random matrix ensembles will provide a widely applicable guide to behavior. Third, the high symmetry of Ginibre s ensemble the matrix elements are independently Gaussian distributed allows for an exact calculation which we present in detail. Separately, we develop an alternative, more general and simpler approach to calculations, based on a perturbative evaluation of ensemble-averaged resolvents, using N as the expansion parameter, where N is the rank of the matrix. We apply this to Girko s ensemble, and also assess its validity in Ginibre s ensemble by comparison with exact results. Such approximate methods are particularly important because they are easily extended to the more general ensembles discussed in Refs. 7 and 8. The remainder of this article is organized as follows. In Sec. II, we discuss the formulation of the problem: we define the ensembles of random non-hermitian matrices studied in subsequent sections, define the densities of eigenvector overlaps that will be the main objects of study in this paper, and the corresponding Green functions. This section establishes the notation used subsequently. In Sec. III, we show how to derive exact results for the statistical properties of eigenvectors in Ginibre s ensemble of non-hermitian random matrices of arbitrary matrix dimensions. We also discuss simplifications which arise in the large N limit. In Sec. IV, we summarize our approximate calculations, applying them to Girko s ensemble. The eigenvector correlators are calculated in terms of two-point functions which are obtained within the self-consistent Born approximation. This approach is appropriate in the limit of large N, and under certain additional assumptions which are discussed in this section. As a special case, we check results found with this method for Ginibre s ensemble against the exact results of Sec. III. The results obtained in Secs. III and IV are summarized and discussed in Sec. V. In Sec. VI, we examine some consequences of eigenvector correlations that are likely to be important in physical applications. These are: the extreme sensitivity of eigenvalues to perturbations; time evolution governed by non- Hermitian random matrices; and the nature of correlations between individual eigenvector components in non-hermitian random matrix ensembles. Finally, we summarize our conclusions in Sec. VII. An outline of some of these results has been published previously in a shorter communication. 22 II. FORMULATION OF THE PROBLEM A. Ensembles of non-hermitian matrices In the recent literature, a large number of different ensembles of non-hermitian random matrices and operators have been discussed. 5 6,3,4 In the following, we restrict ourselves to Ginibre s 3 and Girko s ensembles 4 of non-hermitian random matrices. Ginibre introduced an ensemble of random NN matrices J which have complex elements J kl with independently distributed real and imaginary parts J kl and J kl. The ensemble is defined by the measure 3 see also Ref. PJdJ exp 2 TrJJ kl dj kl dj kl. Thus J kl 0 and J kl J kl 2. Here... denote ensemble averages and the overbar indicates complex conjugation. The parameter 2 controls the density of eigenvalues: in the limit N, the ensemble-averaged density per unit area is / 2 within a disk in the complex plane, centered on the origin and of radius N. Two different conventions are in use for the value of

3 J. Math. Phys., Vol. 4, No. 5, May 2000 Statistical properties of eigenvectors in The choice 2 as for instance, in Refs. and 3 results in a fixed density as N. Alternatively, the choice 2 N results in a fixed support for the eigenvalue density as N. Girko has considered the following generalization of Ginibre s ensemble: PJdJexp 2 2 TrJJ Re JJ k,l dj kl dj kl, 2 with. In this ensemble, the nonzero cumulants are J kl J kl 2, J kl J lk 2. 3 For 0, Ginibre s ensemble is recovered; the case corresponds to Dyson s Gaussian Unitary Ensemble, while describes an ensemble of complex antihermitian matrices. The eigenvalue distribution has been calculated for general in the large N limit by Sommers et al. in Ref. 5: the eigenvalues fill an ellipse in the complex plane with uniform density. Setting 2 N, the real and imaginary axes of the ellipse are and, respectively. The eigenvalue distribution thus collapses onto the real axis for, and onto the imaginary axis for, as it should. B. Densities of left and right eigenvectors The eigenvalues,, and left and right eigenvectors, L and R, of the matrix J satisfy JR R, L J L. 4 In general, the eigenvalues are complex numbers i. Except for a set of measure zero, they are nondegenerate. In this case, the eigenvectors form two complete, biorthogonal basis sets with the normalization L R. 5 The closure relation is L R. 6 We denote the Hermitian conjugates of L and R by L and R, so that, for example, L satisfies J L L. Left and right eigenvectors are generally not orthogonal amongst themselves. On the contrary, scalar products can vary significantly. This can have important physical implications. For instance, it is well known that nonorthogonality of eigenvectors can have an important bearing of time evolution in systems governed by non-normal operators. 2 In the following, we consider statistical properties of scalar products of eigenvectors in ensembles of random non-normal operators. We note that Eqs. 4 and 5 allow for the following scale transformation: R c R, L L c 7 with arbitrary complex numbers c : we study only such combinations of eigenvectors as are invariant under this scale transformation. The simplest such combination of two eigenvectors is trivial see Eq. 5. We hence consider the combination O L L R R. 8

4 3236 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker We calculate the mean value and discuss the distribution function of this overlap matrix. Note that completeness implies the sum rule O. 9 It is convenient to define local averages of diagonal and off-diagonal elements of O, Oz 2 O z, 0 Oz,z 2. 2 O z z 2 Here, zxiy is a complex number with real and imaginary parts x and y and (z) denotes a delta function in both coordinates. Correspondingly, the density of states and the two-point function are defined as, 2 R 2 z,z 2 dz 2 z 2 z z 2 In order to characterize the overlap matrix using Green functions, it is convenient to introduce the density. Dz,z 2 2 O z z 2,, which can be expressed in terms of O(z ) and O(z,z 2 )as 3 4 Dz,z 2 Oz z z 2 Oz,z 2. 5 Thus, information on the diagonal overlap matrix elements may be extracted from the singular part of D(z,z 2 ). The smooth part conveys information on the off-diagonal overlap matrix elements. Finally, we note that the sum rule 9 implies the constraint for the density D(z,z 2 ) d 2 z 2 Dz,z 2 dz, 6 where d(z ) is the density of states Eq. 2. C. Green functions and spectral densities We shall make use of the fact that the densities d(z) and D(z,z 2 ) may be expressed in terms of ensemble averages of resolvents (zj) and products of resolvents (z J) (z 2J ). The density of states d(z), for example, by means of the relation z z z 7 may be expressed as 7,5

5 J. Math. Phys., Vol. 4, No. 5, May 2000 Statistical properties of eigenvectors in dz 2 z TrzJ. 8 In Eqs. 7 and 8, z 2 i. x y Equation 7 replaces the relation (E) Im(Ei0 ) which is applicable in problems for which the Green functions are analytic in the upper and lower complex half-planes. Here, this is not the case. Similarly, the density D(z,z 2 ) can be obtained from a two-point function 9 Dz,z z z 2 Trz J z 2J. 20 This is most easily seen from the spectral representation of the resolvent: zj R z L. We show in Sec. IV how the averages in Eqs. 8 and 20 can be calculated perturbatively, using an expansion in powers of N. III. GINIBRE S ENSEMBLE As pointed out in the introduction, Ginibre s ensemble is a special case of Girko s family of ensembles of non-hermitian matrices. It is obtained by setting 0 in Eq. 2 and is thus the ensemble of complex matrices with independent, Gaussian distributed elements. In this special case, we are able to provide an exact calculation of the eigenvector correlators introduced in Sec. II. A. Density-of-states and eigenvalue correlations Eigenvalue correlations for the ensemble were first studied by Ginibre. 3 The joint probability distribution of the eigenvalues is P N,..., N C N 2 exp 2 2 with normalization C N (N! N j0 j! 2 j2 ). The eigenvalue density and the two-point function d(z) and R 2 (z,z 2 ) Eqs. 2, 3, may be calculated by averaging (z ) and (N)(z )(z 2 2 ) with the weight P N. In the following, we demonstrate briefly a way of performing the corresponding -integrals which can be readily generalized to deal with the integrals that arise in the calculation of the eigenvector correlations Sec. III B 4. Making use of the fact that 2 P N,..., N N! 2N exp 2 2 N m2 m 2 P N 2,..., N, 22 we have dzn 2 N! 2N exp z2 N 2 d d 2 N P N 2,..., N m2 z m This can be written as

6 3238 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker d0 0 d 0 dzn 2 N! 2 exp z2 detd00 2 d N3N2 0 d N2N3 d N2N2 24 with d ij ( j! 2 j4 ) d 2 i j z 2 exp( 2 2 ). Denoting the (N)(N) determinant in Eq. 24 by D N, we derive the recursion relation D k 2 z 2 kd k 2 z 2 kd k. 25 Using D 2 z 2 and D z 2 4 z 4, we thus obtain dz exp z2 2 l0 N z 2l l! 2l 26 which corresponds to Eq in Ref.. In the limit of large N, with 2 N, the density of states is dz Similarly, we obtain for the two-point function with for z, 0 otherwise. R 2 z,z 2 N 2 2 N! 6 z z 2 2 exp 2 z 2 2 z 2 2 f 00 f 0 f 02 0 f 0 f f det f N5N3 f N4N4 f N4N3 0 f N3N5 f N3N4 f N3N3 27 f ij j! 2 j6 d 2 i j z 2 z 2 exp As before, we derive a recursion relation for the (N2)(N2) determinant in Eq. 28. This recursion relation simplifies considerably when z 2 0. Denoting the determinant in Eq. 28 by F N2, we have with z z F k 2 z 2 k2f k 2 z 2 kf k. 30 In this way, we obtain, with F 2 2 z 2 and F z 2 4 z 4 R 2 z,0 2 exp 2 z2 N z 2 l0 2l l! which for 2 is equivalent to in Ref. with n2 and z 2 0. Moreover, in the limit of large N, with 2 N, one finds that for z z 2 and z,z 2, the two-point function is constant 3

7 J. Math. Phys., Vol. 4, No. 5, May 2000 Statistical properties of eigenvectors in... N R 2 z,z 2 2 for z z 2 and z,z 2, 0 otherwise B. Eigenvector correlations In this section, we show how to obtain expressions for correlations of eigenvectors in Ginibre s ensemble. We start from 0 and, perform calculations for general 2 but set 2 N in the final results see Eqs. 68 and 73 below.. Change of basis Since the fluctuations of the eigenvectors and those of the eigenvalues are correlated, it is convenient to parameterize the matrix J following Ref. 23, using a unitary transformation U to bring it into upper triangular form, T2 TN 0 TU JU 2 T 2N. 33 ] ] 0 N The ensemble requires 2N 2 coordinates. Of these, 2N are given by real and imaginary parts of the eigenvalues k, and N(N) by real and imaginary parts of the matrix elements T kl. The remaining N(N) parameters H kl are as described by Mehta. 23 The Jacobian of this transformation is proportional to kl k l 2 and thus depends on,..., N only. Note also that the eigenvector correlator O is invariant under the unitary transformation U. In this section, L and R will denote left and right eigenvectors in the new basis. Thus, TR R and R (,0,...,0) T. In keeping with Eq. 5, let L (,b 2,...,b N ). The coefficients b l can be determined by recursion: From L T T one has, with b and for p, b p p p q b q T qp. 34 The solution of this recursion relation is b, b 2 T 2 2, b 3 T 3 T 2 T , 35 b 4 T 4 4 T 3 T T 2 T T 2 T 23 T , Equation 35 provides an explicit expression for the correlator ] N O l b l 2 36

8 3240 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker in terms of the eigenvalues k and the matrix elements T kl for kl. To calculate off-diagonal correlators one needs, in addition, the eigenvectors L 2 and R 2. Let R 2 (c,,0,...,0) T and, in keeping with Eq. 5, L 2 (0,,d 3,...,d N ). Equation 5 implies that cb 2. Then L 2 T 2 T gives, with d 0 and d 2, d p p 2 p q d q T qp. 37 This recursion relation is solved in the same way as 34 and N O 2 b 2 d lb l l 38 provides a corresponding expression for O 2 in terms of the eigenvalues l and the matrix elements T kl (kl). 2. Integration on T kl It was shown in the previous section how the correlators O and O 2 may be expressed in terms of the eigenvalues k and the matrix elements T kl (kl). The Jacobian depends only on,..., N. In calculating averages of the type 0 and, the N(N) parameters H kl mentioned in Sec. III B can thus be integrated out and only the integrals over T kl for kl and over k for k,...,n remain. These have the form k d 2 k kl k l 2 kl d 2 T kl exp 2 k k 2 2 kl T kl The integrals on all the eigenvalues will be discussed in the next section. In the present section, we show how to perform the integrals over T kl. To this end the notation T is introduced, denoting a normalized integral on all T kl with weight exp( 2 kl T kl 2 ). Consider first the average O T. Let l S l p b p 2 T 40 so that S and S N O T. Then from Eq b l 2 T l 2 S l, 4 and hence S l 2 l 2 S l. 42 Together with S this implies Consider now the average O 2 T. Let N O T l2 2 l l 2. 43

9 J. Math. Phys., Vol. 4, No. 5, May 2000 Statistical properties of eigenvectors in... l S l T 2 k b k d k T so that S 0, S 2 2 /( 2 ) and O 2 T S N /( 2). Now S l S l T 2 b l d l T T 2 l l b l T l q ql k 2 l d kt klt 2 S l. l 2 l 45 This implies 2 O 2 T 2 2 l3 N 2 l 2 l. 46 Equations 43 and 46 represent the averages of O and O 2 with respect to the coordinates T kl. The remaining integrals are those over k. Using Eqs. 43 and 46 one has Oz N 2 z l2 N 2 l 2 P 47 and Oz,z 2 NN 2 z z 2 2 where P is an average with the weight 2. 2 N 2 2 l3 2, l 2 lp The case NÄ2 and The case N2 is particularly simple. We find Oz 2 z2 2 exp z Oz,z 2 2 exp 2 z 2 2 z These expressions are useful as simple checks of results for arbitrary values of N. Of more general interest, the distribution of O is, from Eqs. 34 and 36 PO 4 O 2O 3, 5 where (x) for x0 and zero otherwise. This gives O 3/2 52

10 3242 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker which is consistent with 49 integrated over z. Note that the second and higher moments of O diverge. We argue in Sec. V A 2 that the tail of the distribution of O at large O has the same form for all N2. 4. Calculation of the eigenvalue averages In this section, we show how to evaluate the remaining integrals in 47 and 48. They can be performed in the same way as those in Sec. III A. In analogy with Eq. 24, one has g0 0 g 0 OzN 2 N! 2 exp z2 detg00 2 g N3N2 0 g N2N3 g N2N2 53 with g ij ( j! 2 j4 ) d 2 i j ( 2 z 2 )exp( 2 2 ). Equation 53 provides an explicit expression for O(z) for general N. The determinant can be easily evaluated numerically, as is shown in Sec. V. For z0, the (N)(N) determinant in Eq. 24 is simply diagonal. Denoting it by G N we have G k (k2)g k and thus Oz z0 N, 54 independent of 2. For N2 this expression gives O(0)2/, consistent with Eq. 49. An expression for O(z,z 2 ) can be obtained in analogy with 28: Oz,z 2 N 2 2 N! 4 exp z 2 2 z h0 h02 0 h 0 h deth00 h 20 h N5N3 55 h N4N4 h N4N3 0 h N3N5 h N3N4 h N3N3 with h ij j! 2 j6 d 2 i j z 2 z z z2 exp For z 2 0 and z z, and denoting the determinant in 55 by H N2, we obtain the recursion relation H k 2 z 2 k3h k 2 z 2 k2h k. 57 With H 3 2 z 2 and H z 2 4 z 4 this yields Oz,0N 2 2 exp z2 2 N z 4 l2 z 2l l! 2l. 58

11 J. Math. Phys., Vol. 4, No. 5, May 2000 Statistical properties of eigenvectors in For N2, this gives O(z,0)() 2 exp( 2 z 2 ), which is consistent with 50. An additional check is provided by the fact that Eqs. 54 and 58 obey the sum rule 6. In the limit of N large, with 2 N, we obtain, for z0, Oz,0 2 z 4 59 for z and zero otherwise. In order to exhibit the behavior of Eq. 58 near the origin, for 2 N and in the large N limit, we write N /2 z; for N /2, we then have N 2 Oz, e Equation 60 displays the way in which the result 59 is regularized as z Simplified calculation of the eigenvalue averages for N large The main results of Sec. III B 4 are the determinantal expressions Eqs. 53 and 55, providing exact results for the eigenvector correlators 0 and. In the present section, we provide approximate expressions for 53 and 55 which, for 2 N, are valid in the limit of N large, with z z 2 and z,z 2. In the following, we shall need to indicate explicitly the rank of the random matrix considered, and so we use the notation O N z N O z, 6 O N z,z 2 O N z z 2, 62 in place of O(z) and O(z,z 2 ) Eqs. 0 and. Consider first O N (z). We write O N zo M 0, 63 where N lm N 2 z l 64 and the product excludes the M eigenvalues l closest in the complex plane to the point z, as illustrated in Fig.. We believe that Eq. 63 is exact for N followed by M, because we expect that has no fluctuations in that limit. This implies in particular that we can calculate by evaluating the average of its logarithm. Starting from N log log 65 lm and expanding the logarithm on the right side, we have Nz l 2 log N N lm z l 2 d 2 d D z 2, 66 where in the large N limit, d(z) for z and d(z)0 otherwise. The domain D of integration excludes a disk of radius ϱ with centered on z. Since this disk should contain M eigenvalues, ϱ 2 M/N. Thus, we obtain in the large N limit

12 3244 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker FIG.. Eigenvalue distribution in the complex plane, as discussed in Sec. III B 5. ϱ 2 z Making use of the fact that O M (0)M/ see Eq. 54, and using Eq. 63 we thus obtain Oz N z2. 68 The quantity O N (z,z 2 ) can be calculated in a similar fashion. To this end, we write O N z,z 2 O M z z 2,0 2, 69 where 2 is 2 N lm N z l z 2 l 70 and the product excludes the M eigenvalues closest to z 2, with MN. We first consider the case z z 2. Proceeding as above, we have log 2 d 2 d, D z z 2 7 where again the domain of integration D is the unit disk with a disk of radius ϱ around z 2 removed as illustrated in Fig.. In the large N limit, we obtain 2 ϱ 2 z z As before, ϱ 2 M/N. Using Eqs. 58 and 69, we find in the large N limit and with N /2 (z z 2 ),N /2 N 2 Oz,z 2 z z e Second, we consider the case z z 2 N /2. In this case, we obtain, in the large N limit, for z z 2 and z,z 2,

13 J. Math. Phys., Vol. 4, No. 5, May 2000 Statistical properties of eigenvectors in Oz,z 2 z z 2 2 z z For z,z 2,O(z,z 2 ) vanishes in this limit. IV. GIRKO S ENSEMBLE In this section, we present a general approach to calculating the averages of Eqs. 0 and perturbatively, using an expansion in powers of N. This will enable us to treat more general ensembles than the one considered in the previous section. As an example, expressions are derived for the averages of Eqs. 0 and in the case of Girko s ensemble, defined in Eq. 2. The expressions derived below are appropriate for large N and z z 2 in. For 0, Eqs. 74 and 68 are thus reproduced. In the following, we set N /2 : the results derived are correct in the large N limit. A. Self-consistent Born approximation The desired approximations for 0 and are obtained by calculating the average in Eq. 20 using Green functions. The corresponding Green functions are nonanalytic within the support of the density of states which occupies a finite region in the complex plane. In general, perturbation theory yields only the analytic contribution, and in conventional problems singularities on the real axis are obtained by analytic continuation. In the present case one thus proceeds as follows. A Hermitian 2N2N matrix HH 0 H is introduced: 7,0,4 6,24,25 H 0 A, H A 75 with 0, AzJ and with inverse G 2 AA A 2 A A G G 2 A 2 AA 2 A A G 2 G Expanding the Green function as a power series in H, its ensemble average G can be written as GG 0 G 0 G, 77 where G 0 H 0 and is a self-energy. Within the self-consistent Born approximation, one obtains 7,5 G N 22 zg 2 z G 2 G 78 as illustrated diagrammatically in Fig. 2. The self-consistent Born approximation is exact in the limit N. For 0, the self-consistent solution of Eqs. 77 and 78 is as follows: 5,7,5 one has for all z G 22 G and G 2 Ḡ 2. In addition, G is nonzero only inside the ellipse defined by x/() 2 y/() 2 G x/2 y/ 2 0 outside. inside the ellipse, 79 Furthermore,

14 3246 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker FIG. 2. a Diagrammatic notation for the ensemble 2. b Self-energy in Eq. 77, to lowest order in N, compare Eq. 78. Note that a summation over internal indices in closed loops incurs an additional factor of N. G 2 x/iy/ inside the ellipse, zz 2 4/2 outside. 80 Using Eq. 8, the density of states is given by dz lim z G It thus turns out that for N, the support of d(z) is an ellipse in the complex plane 5,7,5 with dz 2 0 otherwise. for x/ 2 y/ 2, In the limit, the eigenvalue density of the Gaussian Unitary Ensemble is recovered, for which d(z)(y)(2) 4x 2. Alternatively, setting 0, the support of the density of states in the complex plane becomes a disk of unit radius centered around the origin compare Eq B. Bethe Salpeter equation In the following, G kl (z,z ) is denoted by G kl () (k,l,2). An equation for the average of the matrix product G 2 ()G 2 (2), accurate at leading order in N, is shown diagrammatically in Fig. 3. There are sixteen such equations for all products G ij ()G kl (2) for i,...,l,2. In order to write these in matrix form, one defines R,2G Ḡ2. 83 Similarly, R 0 (,2) is the matrix G() Ḡ(2). The matrices R and R 0 are Hermitian. Defining the vertex

15 J. Math. Phys., Vol. 4, No. 5, May 2000 Statistical properties of eigenvectors in FIG. 3. Equation for the matrix product G 2 ()G 2 (2). The diagrammatic rules are as in Fig. 2. N, 84 the diagrammatic expression for R(,2) can be written as R,2R 0,2R 0,2R,2. 85 Equation 86 has the solution R,2R 0,2 R 0,2. 86 We first discuss the simplest case, 0. If z and z 2 lie inside the support of the density of states z and z 2, R 0,2 N In this case, from Eq. 86 z z z z z z z 2z 2 z 2 z 2 z 2 z zz z2z 2 z z 2 2 z z z z 2 R,2 N z 2 z 2 z z 2 z z z 2 z 2z z 2 z z 2 2 z z z z 2 z z z z 2 0 z z 2 z z 2 z z 2 z z 2 0 z 2z 2 z z 2 z z z z 2 z z z 2 z 2z z 2 z z 2 2 z 2z 2 z z 2 z z z z 2 z z z 2 z 2 z z Alternatively, if both z and z 2, we obtain

16 3248 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker z z 2 z z 2 R,2. N 89 z z 2 z z 2 The general case, 0, is dealt with as follows. We define a transformation z w zz 2 90 which maps the support of the density of states in the z plane onto the unit disk in the w plane. For z and z 2 inside the support of the density of states, one has w,w 2 and R 0,2 N w w w w w w w 2w 2 w 2 w 2 w 2 w 2. 9 The resulting matrix R(,2) is more complicated than Eq. 87. For the element G 2 ()G 2 (2) in the case w and w 2, we find G 2 G w w 2 w w w 2 w 2w w 2w w 2 w 2 w 2 2 w w 2 w w C. Calculation of the density D z,z 2 The density D(z,z 2 ) can be expressed in terms of Green functions, from Eq. 20, as Dz,z 2 lim 0 2 z z 2 G 2 G We find from Eq. 92, for z,z 2 within the ellipse, that Dz,z z z 2z 2 z 22 2 z z For z and z 2 outside the ellipse, D(z,z 2 ) vanishes. As a check it can be shown explicitly that D(z,z 2 ) obeys the sum rule 6. Using Green s theorem, we have d 2 z 2 Dz,z 2 dz z 2i 2 G 2 G where the contour integral is around the ellipse. By means of the transformation 90, this contour may be mapped into the unit circle in the w plane, giving

17 J. Math. Phys., Vol. 4, No. 5, May 2000 Statistical properties of eigenvectors in d 2 z 2 Dz,z 2 2 w w 2i w2 dw 2 dw 2G 2 G for w and zero otherwise, as expected from Eq. 82. As a final check, we observe that with 0, Eq. 94 implies Oz,z 2 2 z z 2 z z for z,z 2 and zero otherwise. Thus, our previous result, Eq. 74, is reproduced from 94 for 0. As pointed out in Sec. II B, the diagonal correlator O(z) is given in terms of the singular part of D(z,z 2 ), see Eq. 5. This singular part is inaccessible perturbatively, in lowest order in N. 26 In order to determine O(z) within the perturbative approach discussed in this section, we proceed as follows. For simplicity, consider the case 0. Integrating the density D(z,z 2 ) over a small disk around z 2, of radius which is taken to be small d 2 z 2 Dz,z 2 z z 2 dz 2i 2 z z 2 z G 2 G z 2, provided z is sufficiently far away from the boundary. On the other hand, from Eq. 4 and for N /2, this is approximately O(z ), so that up to prefactors of order O(), 98 Oz Nz 2 99 compare Eq. 68 and thus O N. The sum rule 6 can be used to check the consistency of Eqs. 97 and 99. V. SUMMARY AND DISCUSSION OF THE RESULTS In the present section, we summarize and discuss the results obtained in the previous two sections. As in Sec. IV, the variance 2 in Eqs. and 2 is taken to be /N. A. Ginibre s ensemble. Eigenvector correlators Eqs. (0) and () In the case of Ginibre s ensemble, we have been able to obtain exact expressions for the eigenvector correlators, Eqs. 0 and, in the form of determinants. In certain cases, we could simplify these expressions further by recursion. Combining these results compare Eqs. 54 and 58 with a continuum treatment see Sec. III B 5, in a way which we believe gives exact results for the large N limit, we have for z z 2 0 and z,z 2 N Oz z 2, 00 Oz,z 2 2 z z 2 z z For z,z 2, both densities vanish as N. To display the form of O(z,z 2 )asz z 2 0, it is necessary to express z z 2 in units of the separation between adjacent eigenvalues. Let z (z z 2 )/2, z z z 2, and Nz. For z, N and for N, Eq. 73 implies

18 3250 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker FIG. 4. N O(z)N 2 O (z ) as a function of zr exp i. The ensemble average is independent of. Results for N2,4,8,6 are shown with Eq. 00 valid for large N ---. N 2 Oz,z 2 z e 2 ). 02 We have examined the convergence towards these results for increasing N. In Fig. 4, we show N O(z) as a function of z for N2, 4, 8, and 6, obtained by evaluating the determinant in 53. We also compare this with Eq. 00. The exact results converge rapidly towards the approximate result 00 as N is increased, provided z is sufficiently far from the boundary of the support of O(z). In Fig. 5, we show O(z,z 2 ) as a function of z on the real axis for z for N2, 4, 8, and 6, obtained by evaluating the determinant in 55. We compare this with Eq. 0. Again, the exact results converge rapidly towards the approximate expression 0 as N is increased, provided z and z 2 are not too close to each other or to the boundary of the support of O(z,z 2 ). Finally, in Fig. 6, we show the behavior of N 2 O(z,z 2 ) for z z 2 N /2, comparing the approximate expression 02 with exact results obtained by evaluating the determinant in Eq. 55. The exact results converge very rapidly to the approximate expression as N is increased, provided z and N. It is important to stress the dramatic difference between the behavior of O in Ginibre s ensemble and its behavior in the case of Hermitian matrices, for which O. The fact that, by contrast, O N in the non-hermitian ensemble can be understood as the behavior which would result if L and R were independent random vectors, subject to the normalization of FIG. 5. O(z,z 2 )N O (z 2 )(z 2 ) for z as a function of z on the real axis. Results for N 2,4,8,6 are shown, together with Eq. 0 valid for large N ---.

19 J. Math. Phys., Vol. 4, No. 5, May 2000 Statistical properties of eigenvectors in FIG. 6. Shows Eq. 02 for z 0.4 as a function of for several values of N ---. Also shown are the corresponding exact results. Eq. 5: Choosing a basis and scaling in which R (,0,...,0) T, and assuming that L is a random vector, biorthogonality requires L (,b 2,...,b N ), where the coefficients b j, for j are random and b j is expected to be of order O(). Thus, O N. Moreover, large values for the diagonal elements of the matrix O must be accompanied by some large or many small off-diagonal elements, since the two are linked by the sum rule 9. Indeed, Eq. implies O Oz,z 2 /R 2 z,z 2 03 and hence, from Eq. 02, O N if and are neighboring eigenvalues in the complex plane, so that typically. 2. Distributions of O Finally, it is interesting to ask about, not only the average behavior of the overlap matrix, but also its fluctuations. In fact, O is typically large if the matrix J has an eigenvalue which is almost degenerate with or, and as a result, the probability distribution of O has a power-law tail extending to large O. To illustrate this, we consider N2, for which the probability distribution, P(O ), of a diagonal element of the overlap matrix is given by Eq. 5 and decays at large O according to P(O )O 3. This implies in particular that the second and higher moments of O diverge. For N2, the tail of the distribution P(O ) is determined by pairs of eigenvectors with closest eigenvalues, and we expect that for general N, the tail of the distribution function decays algebraically according to PO O In Fig. 7, we show the distribution P(O ) of the diagonal overlaps O in Ginibre s ensemble for N0. The tail of the distribution function is well described by Eq. 04.

20 3252 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker FIG. 7. A histogram of P(O ) of the diagonal overlaps as a function of O /N in Ginibre s ensemble for N0. Also shown is the theoretical estimate for the tail of the distribution ---. B. Girko s ensemble The main result of Sec. IV is Eq. 94, giving O(z,z 2 ) provided z z 2 is much greater than the mean separation in the complex plane between neighboring eigenvalues. For 0, Girko s ensemble reduces to Ginibre s ensemble. Correspondingly, the perturbative result 94 reproduces, for 0, z z 2 0, z,z 2 and large N the expression 0, which was obtained from the exact results of Sec. III in the same limits. The singular contribution of the diagonal overlap matrix elements to D(z,z 2 ) is only indirectly available within perturbation theory. Equation 99 shows that the singular behavior extracted from the perturbative results is consistent with the exact expressions compare Eq. 00. On the other hand, Eq. 94 implies that for, O O(z,z 2 )/R 2 (z,z 2 ). Thus, O vanishes in the Hermitian limit, as expected. The same is true for the anti-hermitian limit,. VI. IMPLICATIONS Fluctuations of eigenvectors in non-hermitian random matrix ensembles exhibit a number of striking features which are likely to be relevant in physical applications. As in the immediately preceding sections, we take the variance in Eqs. and 2 to be 2 N. A. Sensitivity to perturbations First, as pointed out in the introduction, systems described by a non-hermitian operator are particularly sensitive to perturbations. This sensitivity is determined by the diagonal matrix elements of O. In order to illustrate this fact, it is convenient to consider a one-parameter family of matrices JJ cos J 2 sin, 05 where the parameter is real and the matrices J and J 2 are drawn independently from the same ensemble. Then / 2 N O. 06 According to Eq. 00, O is large, being of order N. Thus, / 2 is of order unity. This should be compared with the Hermitian case, 27 where / 2 is of order N in the corresponding parametrization. Structural stability, on the other hand, requires that the level velocities tend to zero as the boundary of the support of the density of states is approached: the latter must remain unchanged as varies, since the perturbations merely take J from one realization of the ensemble to another. The expression 00 for O(z) shows that this is indeed the case.

21 J. Math. Phys., Vol. 4, No. 5, May 2000 Statistical properties of eigenvectors in B. Time evolution Systems governed by a non-hermitian evolution operator may exhibit transient features in the time dependence of correlation functions which are controlled by the type of correlations between left and right eigenvectors that we have studied. Consider, for example, an evolution equation of the form t u tju t 07 with J drawn from Ginibre s ensemble. We use J rather than J in Eq. 07 for convenience, to suppress exponential growth. This corresponds to shifting the support of the density of states by unity along the negative real axis, so that all except a vanishing fraction of the eigenvalues have negative real parts. Then, u t R f t L u 0, 08 with f t ()exp()t. Ensemble averaging with u 0 u 0 yields u t u t O N f t f t. 09 Thus, properties of the matrix O directly influence time evolution. Equation 09 can be obtained as the double Laplace transform of the density 4, with respect to z and z 2, u t u t d 2 z d 2 z 2 e z z 2 2t Dz,z 2. 0 The diagonal and nondiagonal contributions to D(z,z 2 ) yield large contributions to Eq. 0 which almost cancel. It is thus convenient to evaluate the double Laplace transform in 0 by contour integration, u t u t 2 2 z dz 2e dz z z 2 2t G 2 G 2 2. z2 In this case, one obtains for large N and for tn, u t u t e 2t I 0 2t 2 which, for tn, simplifies to 4t. 3 This behavior should be compared with the much faster decay that would result from the same spectrum if the eigenvectors were orthogonal. In the same regime, the replacement O transforms Eq. 09 into u t u t 2 f N t 4 and thus

22 3254 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker FIG. 8. Shows u t u t as a function of t for Girko s ensemble, for, 0, and. Shown are Eqs. 2 for 0 and7 for. For, one has just u t u t. Also shown are the asymptotic expressions valid for t ---, compare Eq. 3. u t u t t e 2t I 2t 4t 3. 5 Thus, eigenvector correlations may be as significant as eigenvalue distributions in determining evolution at intermediate times, a fact of established importance in hydrodynamic stability theory. 9,20 The more general case of Girko s ensemble 2 can also be treated in this way. Mapping the corresponding contour integrals to the w plane by means of 90, we obtain u t u t e 2t I 0 2tI 2 2t. 6 Here, the support of the spectrum was shifted by along the negative real axis. The limiting cases of Girko s ensemble,, are easily understood: In the anti-hermitian case, for, one has simply u t u t because all eigenvalues have vanishing real parts. In the Gaussian unitary ensemble, for, on the other hand, for large N, d(e)(2) 4E 2 for E2 and thus u t u t 2 de4e 2 e 2E2t 2t e 4t I 2 4t, 7 2 which corresponds to 6 for. For the three cases,, 0, and, u t u t is shown in Fig. 8 full lines together with the corresponding asymptotic expressions dashed lines valid for t. C. Correlations of eigenvector components The space dependence of correlation functions of more general ensembles such as the one discussed in Refs. 7 and 8 can be modeled by correlation functions of the components of L and R. Under a change of basis given by a unitary matrix U, the components of say R transform according to jr jur m U jm mr. Correspondingly, L i L U i l L lū il. Due to the invariance of the ensemble under unitary transformations, we can write L ijr ml U jm Ū il U L lmr, 8

23 J. Math. Phys., Vol. 4, No. 5, May 2000 Statistical properties of eigenvectors in where U denotes an average over the unitary matrices U. With U jm Ū il U N ij lm this implies immediately L ijr N ij. Consider now averages involving four eigenvector components. The only nonvanishing and nontrivial averages which are invariant under the scale transformation 7 are ir L jjl R i N 2 ijo N N 2 ijo 9 and ir L ijl R j N 2 ijo N N 2 ijo. 20 Summing Eqs. 9 and 20 over i and j, one obtains O and unity, respectively, as expected. It should be noted that the dependence on i, j and, does not necessarily factorize. This is likely to be of importance in problems with spatial structure. The above considerations show that interesting space dependence of correlation functions may arise from non-hermiticity. VII. CONCLUSIONS In this paper, we have analyzed correlations of eigenvectors in non-hermitian random matrix ensembles. Such correlations are of interest partly because they determine some aspects of the behavior of systems represented by non-hermitian operators: for example, such systems are particularly sensitive to external perturbations and correlation functions may exhibit transient features in their time dependences. As emphasized in the introduction, there are numerous instances in which random non-hermitian operators appear in the description of physical problems, and we hope that the results and methods summarized here will be of interest in a number of contexts. In particular, we have obtained the following results. We have characterized exactly the eigenvector correlations in Ginibre s ensemble of non-hermitian random matrices. We have shown that the sensitivity of the eigenvalues with respect to external perturbations is larger by a factor of N where N is the rank of the matrix than the equivalent for Dyson s ensembles of Hermitian matrices. Moreover, we have shown that eigenvectors associated with two different eigenvalues exhibit strong correlations which decrease algebraically with increasing separation between the eigenvalues in the complex plane. We have also shown that the probability distribution function of eigenvector overlaps has algebraic tails. This implies that fluctuations are large, in the sense that higher moments of the eigenvector overlaps diverge. In addition to exact calculations specific to Ginibre s ensemble, an alternative, perturbative approach has been developed and used to derive corresponding results, in an approximate way, in Girko s more general ensemble of non-hermitian random matrices. In the appropriate cases and in the limit of large N, the exact results are reproduced. ACKNOWLEDGMENTS One of the authors B.M. gratefully acknowledges support of the SFB393. J.T.C. is supported in part by EPSRC Grant No. GR/MO4426. M. L. Mehta, Random Matrices and the Statistical Theory of Energy Levels Academic, New York, K. B. Efetov, Supersymmetry and Disorder Cambridge University Press, Cambridge, 997; Y. V. Fyodorov and A. D. Mirlin, Int. J. Mod. Phys. B 27, J. Ginibre, J. Math. Phys. 6, V. L. Girko, Theor. Probab. Appl. 29, H. J. Sommers, A. Crisanti, H. Sompolinsky, and Y. Stein, Phys. Rev. Lett. 60, F. Haake et al., Z. Phys. B: Condens. Matter 88, J. T. Chalker and Z. J. Wang, Phys. Rev. E 6, D. R. Nelson and N. M. Shnerb, Phys. Rev. E 58, N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77,

24 3256 J. Math. Phys., Vol. 4, No. 5, May 2000 B. Mehlig and J. T. Chalker 0 K. B. Efetov, Phys. Rev. Lett. 79, K. B. Efetov, Phys. Rev. B 56, I. Y. Goldsheid and B. A. Khoruzhenko, Phys. Rev. Lett. 80, N. Hatano and D. R. Nelson, Phys. Rev. B 58, Y. V. Fyodorov, B. A. Khoruzhenko, and H. J. Sommers, Phys. Rev. Lett. 79, ; Y. V. Fyodorov, H. J. Sommers, and B. A. Khoruzhenko, Ann. Inst. Henri Poincaré 68, R. A. Janik et al., Phys. Rev. E 55, J. Feinberg and A. Zee, Nucl. Phys. B 504, N. Lehmann and H. J. Sommers, Phys. Rev. Lett. 67, C. Mudry, B. D. Simons, and A. Altland, Phys. Rev. Lett. 80, L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, Science 26, B. F. Farrel and P. J. Ioannou, J. Atmos. Sci. 53, L. N. Trefethen, SIAM Rev. 39, J. T. Chalker and B. Mehlig, Phys. Rev. Lett. 8, Appendix 35 of Ref.. 24 R. A. Janik, M. A. Nowak, G. Papp, and I. Zahed, hep-ph/ unpublished. 25 R. A. Janik, M. A. Nowak, G. Papp, and I. Zahed, Nucl. Phys. B 498, Very recently, a direct method has been proposed for calculating O(z) within a perturbative expansion: see R. A. Janik, W. Noerenberg, M. A. Nowak, G. Papp, and I. Zahed, Phys. Rev. E 60, M. Wilkinson, J. Phys. A 22,