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Finite-difference distributions for the Ginibre ensemble

The Ginibre ensemble of complex random matrices is studied. The complex valued random variable of second difference of complex energy levels is defined. For the N=3 dimensional ensemble are calculated distributions of second difference, of real and imaginary parts of second difference, as well as of its radius and of its argument (angle). For the generic N-dimensional Ginibre ensemble an exact analytical formula for second difference's distribution is derived. The comparison with real valued random variable of second difference of adjacent real valued energy levels for Gaussian orthogonal, unitary, and symplectic, ensemble of random matrices as well as for Poisson ensemble is provided.


I. INTRODUCTION
Random Matrix theory assumes that the Hamiltonian operator H of a generic quantum system is unknown and unknowable [1][2][3]. The matrix elements H ij of Hamiltonian in given basis of Hilbert space are random variables. Their distributions are given by appropriate formulae depending on studied Random Matrix ensemble [1][2][3]. Thy symmetry properties of H which is hermitean lead us to Gaussian ensembles of random matrices: orthogonal GOE, unitary GUE, symplectic GSE, as well as to circular ensembles: orthogonal COE, unitary CUE, and symplectic CSE. The energies E i of quantum systems calculated from diagonalization of Hamiltonian matrix H ij are random variables with appropriate distributions and they exhibit generic classes of level repulsion. It was Wigner who firstly discovered level repulsion phenomenon [1][2][3]. The applications of Random Matrix theory are very broad: nuclear physics (slow neutron resonances, highly excited complex nuclei), condensed phase physics (fine metallic particles, random Ising model [spin glasses]), quantum chaos (quantum billiards, quantum dots), disordered mesoscopic systems (transport phenomena). J. Ginibre studied very general case of random Hamiltonians. He dropped the assumption of hermiticity of Hamiltonians and he considered generic complex valued matrices [1,2,4,5]. Thus, H's belong to general linear Lie group GL(N, C), where N is dimension and C is complex numbers field. Therefore, the energies Z i of quantum system ascribed to Ginibre ensemble are complex valued. This is an extension of Gaussian or circular ensembles. J. Ginibre postulated the following joint probability density function of random vector of complex eigenvalues Z 1 , ..., Z N for N × N Hamiltonian matrices [1,2,4,5]: We emphasise that Z i 's are complex valued random variables, and z i 's are complex sample points (z i ∈ C). One must emphasise here Wigner and Dyson's electrostatic analogy. A Coulomb gas of N unit charges moving on complex plane (Gauss' plane) C is considered. The vectors of positions of charges are z i 's and potential energy of the system is: If gas is in thermodynamical equilibrium at temperature T = 1 2k B (β = 1 k B T = 2, k B is Boltzmann's constant), then probability density function of vectors of positions is P (z 1 , ..., z N ) Eq. (1). Thus, complex energies of quantum system and vectors of positions of charges of Coulomb gas are analogous to each other. In view of above analogy one can consider the complex spacings ∆ 1 Z i of complex energies of quantum system: as vectors of relative positions of electric charges of Coulomb gas. For Ginibre ensemble it were calculated the distributions of real valued absolute values of spacings of nearest neighbour ordered energies. We complement this by introduction of complex valued second differences ∆ 2 Z i of complex energies: The second differences are three energy level magnitudes that enhance our knowledge of quantum systems with complex energies. Moreover, ∆ 2 Z i 's can be regarded as vectors of relative positions of vectors of relative positions of electric charges. One can observe movement of electric charges is Cartesian frame of references or in polar one. Since the twodimensional vectors in Cartesian frame of reference have their projections on co-ordinate axes, therefore the real and imaginary parts of ∆ 2 Z i 's, namely Re∆ 2 Z i , Im∆ 2 Z i , can be interpreted as projections of second differences on abscissa and ordinate axes, respectively. The radii |∆ 2 Z i |, and arguments (angles) Arg∆ 2 Z i of second differences have interpretations of polar co-ordinates of ∆ 2 Z i vectors. ∆ 2 Z i 's are analogous to real valued second differences: of adjacent ordered increasingly real valued energies E i defined for GOE, GUE, GSE, and Poisson ensemble PE (where Poisson ensemble is composed of uncorrelated randomly distributed energies) [6][7][8][9]. We will calculate the distributions of Finally, we will compare these results with second difference distributions for Gaussian ensembles, and Poisson ensemble [1,[10][11][12][13][14][15].

II. SECOND DIFFERENCE DISTRIBUTIONS
We use formula (1) with N = 3 and define the following complex valued random vector (Y 1 , Y 2 , Y 3 ) and real A j and imaginary B j parts: The change of the variable formula gives us the result for joint probability density function of random vector (Y 1 , Y 2 , Y 3 ) [16]: where y j = (a j , b j ) ∈ C are complex random sample points. [In order to obtain Eq. (7) we used the complex valued linear map where Jacobian of the inverse map Ξ −1 is equal to unity Jac(Ξ −1 ) = 1]. We integrate out Y 3 : and we obtain following marginal probability density function: Now we calculate the marginal probability density function of second difference: Now we derive the distributions of real part A 1 and of imaginary part B 1 of second difference: where R is field of real numbers. We transform complex valued random variable of second difference Y 1 to polar coordinates' variables R 1 , Φ 1 : and we obtain by standard method the following probability density function of random vector (R 1 , Φ 1 ): (the Jacobian of transformation is r 1 , Θ is Heaviside (step) function [16]). It follows that:

III. N-DIMENSIONAL GINIBRE ENSEMBLE
The case of generic N-dimensional Ginibre ensemble is of special physical interest (N ≥ 3). We will calculate the distribution of second difference for the ensemble. One commences with n-level correlation function: Ref. [1]. In order to calculate the distribution of complex second difference W 1 = ∆ 2 Z 1 for N-dimensional Ginibre ensemble one substitutes n=3 to Eq. (17) and defines random vector The probability density function of random vector W reads: Ref. [16]. Hence, the distribution of second difference is: We combine Eqs (17) where P is permutation of indices (1,2,3). The only nonzero contribution to Eq. (23) is for identity permutation P = id = (1, 2, 3). It results from the fact that other permutations produce factors that are periodic functions of arguments Argw 2 , Argw 3 of complex numbers w 2 , w 3 (the integrals over w 2 , w 3 can be transformed to polar co-ordinates where the arguments Argw 2 , Argw 3 are integrated over [0, 2π]). Hence, Therefore, considering Eq. (19) one has: One changes variables in Eq. (26) in following way: V 2 = 2W 2 , V 3 = −W 3 , and obtains: The above double integral can be calculated by extending the exponent by additional terms proportional to λ i parameters and considering appropriate derivatives: Finally, we derive F (w 1 , λ 1 , λ 2 , λ 3 ) by transformation of parametric quadratic form G(w 1 , v 2 , v 3 , λ 1 , λ 2 , λ 3 ) to canonical form and integrating over v 2 , v 3 : where Hence, we obtained analytical formula for distribution P 3 (w 1 ) of second difference for Ndimensional Ginibre ensemble combining Eqs. (25), (28), (31), (32), (33). What is worth to be mentioned is that second difference's distribution is a triple sum of zero-centred Gaussian distributions with different widths. The distribution is again function of only modulus |w 1 | of second difference and it has global maximum at origin.

IV. COMPARISON
Finally, in order to compare our results for second difference for Ginibre ensemble with previous ones for Gaussian and Poissonian ensembles we must rescale them by division by appropriate magnitude. We consider such rescaled dimensionless second differences in following way. The mean values of second differences either in real or in complex case are zero, hence we cannot divide second differences by the mean values. It follows that we divide real valued second differences ∆ 2 E 1 for GOE(3) (β = 1), for GUE(3) (β = 2), for GSE(3) (β = 4), for PE (β = 0), by mean spacings < S β > calculated for those ensembles, and we create new dimensionless second differences: respectively [6][7][8][9]. The probability density functions of C β 's were calculated for GOE, GUE, GSE, and PE [6][7][8][9]. Since second difference for Ginibre ensemble is complex valued, then we choose its real part A 1 for comparison with C β 's. One divides A 1 by analogue of < S β >'s, which is mean value < R 1 > of radius R 1 Eq. (16) of ∆ 2 Z 1 : Hence: is rescaled dimensionless A 1 . The probability distributions of C β 's and of X 1 depend on the same real variable x which is equal to ∆ 2 e 1 <S β > , a 1 <R 1 > , respectively (e 1 is value of energy E 1 ). The second differences' distributions for Gaussian, Poisson, and Ginibre ensembles assume global maxima at origin and that they are unimodular. Firstly, it extends the theorem of level homogenisation to Ginibre ensemble [6][7][8][9]. We can formulate the following law: Energy levels for Gaussian ensembles, for Poisson ensemble, and for Ginibre ensemble tend to be homogeneously distributed. The second differences' distributions assume global maxima at origin no matter whether second differences are real or complex. From Coulomb gas' point of view it is easier to be understood. The unit charges behave in such a way that the vectors of relative positions of vectors of relative positions of charges statistically tend to be zero. It could be called stabilisation of structure of system of electric charges. The above results can be extended to study of higher differences' distributions for Ginibre ensemble.

V. ACKNOWLEDGEMENTS
It is my pleasure to deepestly thank Professor Jakub Zakrzewski for formulating the problem.