Eigenvalue processes of Elliptic Ginibre Ensemble and their Overlaps

We consider the non-hermitian matrix-valued process of Elliptic Ginibre ensemble. This model includes Dyson's Brownian motion model and the time evolution model of Ginibre ensemble by using hermiticity parameter. We show the complex eigenvalue processes satisfy the stochastic differential equations which are very similar to Dyson's model and give an explicit form of overlap correlations. As a corollary, in the case of 2-by-2 matrix, we also mention the relation between the diagonal overlap, which is the speed of eigenvalues, and the distance of the two eigenvalues.


Introduction
In random matrix theory, the study of eigenvalue processes was started by Dyson [11]. He considered the hermitian matrix-valued process whose entries are given by independent Brownian motions and derived the stochastic differential equations of the eigenvalues by perturbation theory: dt , i = 1, · · · , N (1.1) where β = 1, 2, 4. The parameter β implies the matrix symmetry and corresponds to Gaussian orthogonal, unitary and symplectic ensembles (GOE, GUE and GSE) as β = 1, 2, 4, respectively [3,26]. These processes are called Dyson's Brownian motion model, and when β = 2, we call simply them Dyson's model in this paper. For β ≥ 1, the processes satisfying (1.1) are noncolliding [27], and so are the above three eigenvalue processes. This fact is naturally expected because the eigenvalues exert "repulsive" force on each other [29]. For other studies of time dependent random matrices, Dyson also derived the eigenvalue processes of unitary matrices in the same paper [11], and Bru derived that of the positive definite hermitian matrix which is called Wishart processes [6,7]. Moreover, the relation between eigenvalue processes and noncolliding diffusion particle systems was reported in [17,[21][22][23]. We remark that all the above matrices are normal and their eigenvalue processes are diffusion.
The aim of this paper is to derive the eigenvalue processes of Elliptic Ginibre ensemble (EGE) and show the relation between Dyson's model and the time evolution model of Ginibre ensemble with overlaps. EGE is one of the non-hermitian random matrix models, and recently, many applications of non-hermitian random matrices are discussed in physics: for example, resonance scattering of quantum waves in open chaotic systems, quantum chromodynamics at non-zero chemical potential [1] and neural network dynamics [10]. We note that the matrix model used in the third example is very similar to EGE. To return to our subject, we begin to explain the statistic result for EGE and overlaps in random matrix theory. EGE was introduced as an interpolation between hermitian and non-hermitian matrices by J : [28]. Here, H 1 and H 2 are independent GUE, that is, distributed in the hermitian matrix space in R N 2 with density p(H) ∝ exp[−N tr(H 2 )]. The parameter τ implies the degree of hermiticity. With τ = 1, the matrix J is hermitian and thus GUE, and with τ = 0, it is completely non-hermitian and thus Ginibre ensemble [15,26]. For −1 < τ < 1, J is distributed in the complex matrix space C N 2 with density p(J) ∝ exp − N 1−τ 2 tr(JJ * − τ 2 (J 2 + J * 2 )) . The joint probability density function (jpdf for short) of the eigenvalues of J is described explicitly as in [1,24]. For fixed −1 < τ < 1, the limiting empirical spectral distribution converges to the uniform distribution on the ellipse z ∈ C ; Re(z) 1+τ 2 + Im(z) 1−τ 2 ≤ 1 in [28]. This convergence is known as "elliptic law" [16]. The condition −1 < τ < 1 is called "strong non-hermiticity" since the anti-hermitian part √ −1 √ 1 − τ H 2 of J is the same order in N as the hermitian part. In contrast, the limiting behavior under the condition of "weak non-hermiticity" is also known. Suppose that 1 − τ = α N by some α > 0. Then in the limit N → ∞, the density of the eigenvalues z = x + √ −1y of J behaves asymptotically as p sc (x)p(y), where p sc (x) = 1 2π √ 4 − x 2 on the interval [−2, 2] (Wigner's semicircle distribution) and p(y) = 1 √ 2πα exp(− N 2 y 2 2α 2 ) (Gaussian distribution) . This result is first observed in [13] by perturbation theory, but nowadays there are more detailed studies by using correlation function and kernels [1,2].
For non-normal matrices, the overlaps also have been studied. They are also called eigenvector correlations or condition numbers. For the right eigenvectors R j and left eigenvectors L j , j = 1, · · · , N , the overlaps are defined by For normal matrices, overlaps are trivial, that is, O ij ≡ δ ij ; on the contrary, for non-normal cases, they play an important role because the non-orthogonality of eigenvectors effects the behavior of eigenvalues [30]. In the case of Ginibre ensemble, an early observation was given by Chalker and Mehlig. They estimated the asymptotic behavior of the conditional expectation E[O 11 |λ 1 = z] ∼ N (1−|z| 2 ) as N → ∞ in [9]. Recently, Bourgade and Dubach showed the limiting conditional distribution of O 11 N converges to inverse gamma distribution for complex Ginibre ensemble in [5] with probabilistic approach, and Fyodorov also showed a similar result for real Ginibre ensemble with supersymmetric approach in [12]. Furthermore, the result for real EGE was also reported in [14].
The study of matrix-valued process for non-normal matrices is lesser than that for normal case since the eigenvectors and overlaps should be also concerned together as mentioned above. Nevertheless, there are remarkable results for Ginibre ensemble. This model is non-symmetric matrix-valued process whose entries are given by independent complex Brownian motions. Grela and Warcho l solved the Fokker-Plank equation of the jpdf of the eigenvalues and eigenvectors [8,18] and observed the correlation between the distance of two eigenvalues and the diagonal overlap by numerical experiment with cooperators [4]. Bourgade and Dubach mentioned in [5] that the complex quadratic variations of the eigenvalue processes are truly the overlaps, and they also suggested the limiting behavior of the eigenvalues as the matrix size goes to infinity. In the above two results, they also pointed out that the stochastic differential equations of the eigenvalue processes have no drift term, without using their explicit forms; the eigenvalue processes are complex martingales. This is an unexpected and counterintuitive fact because eigenvalues should exert "repulsive" force on each other by random matrix theory, and this force appear in the drift terms for normal matrix cases as in (1.1). For this reason, overlaps are very important quantities to understand the behavior of eigenvalue processes well.
The main result of this paper is to give the stochastic differential equations (2.5), with parameter −1 ≤ τ ≤ 1, that the eigenvalue processes of EGE satisfy. On the basis of the above results and observations, we consider the non-normal matrix valued process of EGE whose entries are given by independent Brownian motions. This model naturally gives an interpolation between Dyson's model and the time evolution of Ginibre ensemble by using hermiticity parameter τ in the same way as the statistic case. In the main theorem (Theorem 2.1), we show that for −1 ≤ τ ≤ 1, the eigenvalue processes of EGE satisfy the stochastic differential equations which have the drift of Dyson's model except for τ = 0 and also show the explicit form of their timedepending overlaps described by given Brownian motions. As a result, we obtain the complex martingales of the eigenvalue processes of Ginibre ensemble explicitly, which tell us the interaction of the eigenvalues by the form of difference product, that is, Vandermonde determinant. In the case of 2 × 2 matrix, we can show that the quadratic variation of the diagonal overlap and the distance of the two eigenvalues is negative by using our explicit forms, which proves the fact in [4]; as the two eigenvalues get closer to each other, they move faster. We also show for −1 ≤ τ ≤ 1, the eigenvalue processes of EGE are non-colliding.
The organization of the paper is the following. In section 2, we show our main theorem and corollaries with some observations. In section 3, we give the proof of Theorem 2.1 and Corollary 2.5. We put together some properties of characteristic polynomials, eigenvalues and determinants in Appendix.

Settings and Main Results
We consider the N × N matrix-valued process for EGE and define this model as follows: Here, are independent one-dimensional Brownian motions defined on a filtered probability space (Ω, F, {F t } t≥0 , P). The entries of J(t) have the correlation described by the complex quadratic variations: Here, for complex semi-martingales M (t) = M R (t)+ √ −1M I (t) and N (t) = N R (t)+ √ −1N I (t), the complex quadratic variation is defined as The quantities (2.2) express the hermiticity of the matrix J(t) by τ . By construction of J(t), we get Dyson's model with τ = 1 and Ginibre dynamics with τ = 0. J(t) has pure imaginary eigenvalue processes with τ = −1, which are just Dyson's model on the imaginary axis. Thus it is essential to consider 0 ≤ τ ≤ 1. From the perspective of normality of matrix, the case of τ = 1 and τ = 0 are extreme; in the former case each of the eigenvalue processes has the drift term, similar to that in (1.1), which takes a larger absolute values as it gets closer to the other eigenvalues, and in the latter case the eigenvalue processes are complex martingale. We denote the eigenvalue processes of J(t) by λ λ λ(t) = (λ 1 (t), · · · , λ N (t)). As mentioned above, these processes usually take complex values, so we write λ i (t) = λ R i (t) + √ −1λ I i (t), i = 1, · · · , N . We assume the following initial condition: We denote the N × N identity matrix by I, and for a square matrix A, we define the (N − 1) × (N − 1) minor matrix A k| that is obtained by removing the k-th row and the -th column from A. Then we have the following result.

(2.8)
Moreover, the eigenvalues do not collide each other, that is, By (2.6) and (2.7), the complex quadratic variations of λ λ λ(t) are described as d λ i , λ j t = O ij (t)dt. Although we do not use non-orthogonality of the eigenvectors in our proof, the quadratic variations coincide with the overlaps of J(t) because the result and proof for the eigenvalue processes of Ginibre ensemble in [5] are also valid for our model. For this reason, we use the notation O ij (t) for the quadratic variations.
Indeed, the overlaps (2.8) are sensitive for the normality of matrices as shown below.
Proof. J(t) is hermitian with τ = 1, and so each of the matrices the matrix has only one zero eigenvalue, and for the numerator of (2.8), by Lemma A.2 we obtain On the other hand, for i = j, the matrix has two zero eigenvalues. Hence this summation vanishes.
The parameter τ implies the hermiticity and controls the speeds of the real and imaginary parts of λ i (t). Theorem 2.1 shows that for τ = 0, the drift term of λ i (t) completely vanishes, and this fact is observed in the previous study of Ginibre ensemble. Moreover, (2.6) states that each trajectory of the eigenvalue processes of Ginibre ensemble is Brownian motion, whereas they never collide.
Corollary 2.4. Only for τ = 0, each of the eigenvalue processes is conformal martingale. Hence for each i = 1, · · · , N , let In the case of the matrix size N = 2, the numerical experiment of the relation between the distance of the two eigenvalues |λ 1 (t)−λ 2 (t)| and the diagonal overlap O 11 (t) was observed in [4]. They reported that O 11 (t), which is the speed of the eigenvalue processes, takes a larger value as the two eigenvalue processes get closer to each other. We attempt to justify this observation as follows. By (2.8), we have the explicit forms of the overlaps in N = 2: The following relations hold: We note that O 11 (t) and O 12 (t) are real valued process for N = 2, nevertheless O ij (t) are complex valued process for N ≥ 3. By (2.11), we need only to consider the diagonal overlap O 11 (t).
Corollary 2.5. For N = 2 and −1 < τ < 1, O 11 (t) satisfies the stochastic differential equation: Remark 2.6. For τ = 1 and τ = −1, J(t) is normal and O 11 (t) ≡ 1, so that the negative correlation between O 11 (t) and |λ 1 (t) − λ 2 (t)| 2 vanishes. In particular, for the deterministic parameter τ , O 11 (t) takes the maximum value at τ = 0. To show this, we simply deal with the case of the initial condition that each of the eigenvalues starts at the origin, or equivalently, statistic EGE whose entries are given by independent centered gaussians with variance t. By schur decomposition, there exists a unitary matrix U (t) such that where X is a complex Gaussian with mean 0 and variance t. Using this, we have where Y is independent of λ 1 (t), λ 2 (t) and obeys exponential distribution with parameter 1. We notice that a similar deformation is obtained in [5] for Ginibre ensemble. Consequently, the complete non-normality at τ = 0 provides the biggest negative correlation of O 11 (t) and |λ 1 (t) − λ 2 (t)| 2 and effects the behavior of the two eigenvalues significantly.

Proof of Theorem 2.1
Firstly, we derive the stochastic differential equations (2.5) by implicit function theorem until the first collide time t ∈ [0, T col ), and finally we show T col = ∞, a.s. The detail calculations are summarized in subsection 3.2. We define the N × N deteriministic matrix J as where H 1 , H 2 are hermitian: Here, We denote f = f R + √ −1f I and the partial derivative of f with respect to η by f η and also define f R := ∂f ∂λ R and f I := ∂f ∂λ I . Assume that J has simple spectrum with the eigenvalues λ 1 , · · · , λ N . f is analytic with respect to λ, and so for all i = 1, · · · , N , the Jacovian is non-zero: Hence we can apply implicit function theorem for each λ i , and we obtain the derivative of λ i by using that of f as follows: By using (3.1), we can also calculate the second derivatives of λ R i and λ I i which give us the drift terms in the stochastic differential equations (2.5) and the quadratic variations in (2.6) and (2.7). For a C 2 function g : R N 2 → R, we define the gradient and Laplacian of g by ∇g := ∂g ∂x 11 , ∂g ∂x 12 , · · · , ∂g ∂x N N , ∂g ∂α 11 , ∂g ∂α 12 , · · · , ∂g ∂α N N , ∂g ∂y 12 · · · , ∂g ∂y N −1N , ∂g ∂β 12 · · · , ∂g ∂β N −1N , In the notation, a key lemma holds as follows.
Lemma 3.1. For i = 1, · · · , N , λ R i and λ I i are C 2 function. We have Moreover, the inner products of the gradients of λ R i and λ I i are following: We give the proof of Lemma 3.1 in subsection 3.2. As a result, we can derive the stochastic differential equations (2.5) and the quadratic variations (2.6)-(2.8). Under the assumption that J has simple spectrum, we are able to use the above calculus and apply Ito's formula for λ R i and λ I i until first collision time T col defined as (2.9). Up to the time T col , we have For the local martingale part of (3.5), we have f λ (λ i ) by (3.1). Using (3.17), (3.18) and (3.20) in the next subsection, we find Therefore, the local martingale part of (3.5) is For the drift part of (3.5), by (3.2) and Lemma A.1, we have . (3.7) From (3.5)-(3.7), we obtain the stochastic differential equations (2.5) for t ∈ [0, T col ). Next, we derive the quadratic variations (2.6)-(2.8). Because the local martingale part of λ R i and λ I i are constructed by 2N 2 independent Brownian motions as (3.5), we immediately find that  Proof. Rogers and Shi showed that collision time is infinity almost surely for general stochastic differential equations which include Dyson's model [27]. Hence we have only to show the claim for −1 < τ < 1. However, their method does not work for our complex eigenvalue processes straightforward because the martingale terms have the correlations (2.6)-(2.8). Accordingly, we refer to the method in [6,25]. Assume that T col < ∞ and define U : C N → C as follows:

By Lemma A.4, we rewrite the summation of the numerator of (3.3) as
U is an analytic function with respect to z i , i = 1, · · · , N and the derivatives of U are From (2.6) and (2.7), we obtain the complex quadratic variations of the eigenvalue processes as We denote dλ dt, i = 1, · · · , N . Applying Lemma A.5 to U and the eigenvalue processes for t ∈ [0, T col ) with (3.9), we obtain dt.
Indeed, the last summation vanishes. Proof.
t , t ∈ [0, T col ), and so U (λ λ λ t ) is a complex local martingale. We rewrite U (λ λ λ t ) = U R t + √ −1U I t . In the limit of t → T col , by definition of U the radial part diverges. Hence either one of the divergence holds: In the former case, U R t is an one-dimensional local martingale whose real quadratic variation is We define T R (t) := inf{u ≥ 0; U R u > t}, and so the time changed process is an one-dimensional Brownian motion in the usual manner. Hence which never occur by the properties of Brownian motion's paths [20]. In the latter case, we also have the same contradiction by applying time change to U I t with the real quadratic variation Therefore, we have the contradiction in both cases, and the claim holds.
Together with Proposition 3.2, we complete the proof of Theorem 2.1.

Proof of Lemma 3.1
To show Lemma 3.1, we need to calculate the first and second derivatives of λ R i and λ I i which are described by those of f as a result of implicit function theorem. For η = x k , α k , y k , β k , we apply chain rule to the first derivatives of λ R i and λ I i in (3.1) and obtain Taking the summation in (3.10) and (3.11) for η = x k , α k , y k , β k , we yield (3.13) From (3.1), we also have the gradient terms of λ R i and λ I i : (3.14) To calculate the above quantities, we must know the derivatives of f explicitly, and so we use Lemma A.3 with A = λI − J. We note that for k < , each determinant in (A.3) does not have (k, k) and (k, ) entries, and we obtain For the off-diagonal entries of A, Expanding det (λI − J) k| and det (λI − J) |k by each the -th and k-th row, we have and we yield Similarly, we also have the other first and second derivatives of f : (3.20) Using these derivatives and applying Lemma A.4, we get Indeed, the summation of (3.21) has an useful expression.
Therefore, we finish the proof of Corollary 2.5.
Proof. Applying binomial expansion to the characteristic polynomial f (λ) = det(λI N − A), we have We also apply Fredholm determinant expansion to f (λ) and obtain Therefore, the claim holds by comparing the coefficient of λ N −k each other. where A k |pq is the (N − 2) × (N − 2) minor matrix that is obtained by removing the k, -th rows and the p, q-th columns from A.
Proof. We expand det A by the k-th row, and we also expand each of the (N −1)-th determinants by the -th column.