Density of small singular values of the shifted real Ginibre ensemble

We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter $z$ as the dimension tends to infinity. For $z$ away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in [arXiv:1908.01653]. On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter $z$ becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula [arXiv:0707.2929] in a regime where the main contribution comes from a three dimensional saddle manifold.


A
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We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter z as the dimension tends to infinity. For z away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in [ ]. On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter z becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula [ ] in a regime where the main contribution comes from a three dimensional saddle manifold.
. I The universality paradigm in random matrix theory asserts that the local eigenvalue statistics of large random matrices depend only on the basic symmetry class of the ensemble. In the Hermitian case, this dependence is usually investigated for the k-point functions starting from k ≥ 2, while the one-point function is largely insensitive to the symmetry class apart from finite-size correction terms (see, e.g. [ ] for GOE/GUE). For non-Hermitian matrices, however, the real axis plays an interesting distinguishing role between the real and complex ensembles already on the level of the one-point function. In the simplest Gaussian case this phenomenon has been well known for the eigenvalues; in this paper we investigate it for singular values where no explicit formulas are available.
We consider the real or complex Ginibre ensemble [ ], i.e. large N × N random matrices X with independent identically distributed (i.i.d) real or complex Gaussian entries x ab . The customary normalization, E x ab = 0, E |x ab | 2 = N −1 , guarantees that the density of eigenvalues of X converges to the uniform measure on the complex unit disk z |z| ≤ 1 , known as the circular law, and that the spectral radius of X converges to with very high probability (these results also hold for general non-Gaussian matrix elements, see e.g. [ , , , , , , ]).
While the distribution of the complex Ginibre eigenvalues is clearly rotationally invariant, the real axis plays a special role for the real Ginibre ensemble, in particular there are typically ∼ √ N real eigenvalues [ ] (see also the exact formula for having precisely k real eigenvalues in [ ]). In fact, all correlation functions of the Ginibre eigenvalues are explicitly known see [ ] and [ ] for the simpler complex case, and [ , , , ] for the more involved real case. The precise formulas reveal a remarkable phenomenon [ , Theorem ]: the local eigenvalue statistics for real Ginibre matrices coincide with those for complex Ginibre matrices anywhere in the spectrum away from the real axis (see also [ ]).
To what extent does this phenomenon hold for low lying singular values of X and their shifted version X − z with a complex parameter z? While singular values may behave very differently than eigenvalues, intuitively the very small singular values of X − z are still related to the eigenvalues of X near z, since z is an eigenvalue of X if and only if X − z has a zero singular value. Hence we expect that these small singular values of X − z for z away from the real axis behave in the same way for real and complex Ginibre matrices. This was recently proven in [ , Theorem . ] for all k-point correlation functions and even for any i.i.d. (i.e. not necessarily Gaussian) distributed matrix elements but only in the regime | z| ∼ 1. In this paper we prove that this phenomenon holds down to very close to the imaginary axis, | z| N −1/2 , on the level of the density (or one-point function) of the singular values using supersymmetric (SUSY) techniques. In our related paper [ ] we explore the power of this approach with applications to numerical analysis by establishing new bounds on the eigenvector condition number and on the eigenvector overlaps [ , , ].
More precisely, we find that in the large N limit the density of the low lying singular values of X −z for a real Ginibre matrix coincides with that of the complex Ginibre matrix X as long as | z| N −1/2 , while it is different for | z| ∼ N −1/2 , c.f. Figure . This indicates a transition in the local Histogram of rescaled smallest singular value N σmin(X − z) of X − z in the real (dark grey) and complex (light grey case). For z = 0 the asymptotic densities 2xe −x 2 and (1 + x)e −x 2 /2−x have been computed by Edelman [ ].
singular value statistics of X − z from real to complex as | z| increases beyond N −1/2 , similarly to the local eigenvalue statistics of X. Technically, we express the averaged trace of the resolvent of (X − z)(X − z) * in terms of contour integrals using the superbosonization formula [ ] and perform the large N limit. This analysis has been carried out for the complex case in [ ], now we handle the considerably more involved real case. The main additional complication stems from the structure of the superbosonization formula: the contour integration in the real case involves three integration variables, two of them are highly convolved and their contours cannot be deformed independently; while the complex case has only two variables and the phase function is decoupled in them. The entire analysis is done at the bottom of the spectrum of (X − z)(X − z) * , at a distance comparable with the (square of the) local spacing of the singular values, hence our result directly gives precise information on individual singular values. In this critical regime the answer does not come simply from a saddle point, but from a genuine three-fold integral even after the N → ∞ limit is taken. With a careful choice of the interdependent deformations of the contours we achieve the negative sign in the real part of the phase function hence we can rigorously estimate the physically irrelevant highly oscillatory integration regimes. Note that the mere existence of such deformation is not guaranteed by any physical principle, let alone finding them explicitly -this is what we achieve here. A further feature of our work is that we can handle the bulk, |z| < 1, as well as the edge regime, |z| ≈ 1, where the scaling changes from N −1 to N −3/4 .

Notations and conventions.
For positive quantities f, g we write f g and f ∼ g if f ≤ Cg or cg ≤ f ≤ Cg, respectively, for some constants c, C > 0 which are independent of the basic parameters of the problem N, λ, η, δ in ( ). For any two positive, possibly N -dependent, quantities f, g we write f g to denote that f N − g, for some small > 0 (however this convention will We remark that [ , Theorem ] did not explicitly state that the transition takes place for | z| N −1/2 , but it can be concluded from its proof. be locally altered within the proof of Lemma . ). We abbreviate the minimum and maximum of real numbers by a ∧ b := min{a, b} and a ∨ b := max{a, b}.

. M
We consider the ensemble Y z := (X − z)(X − z) * with X ∈ R N ×N being a real Ginibre matrix, i.e. its entries x ab are such that √ N x ab are i.i.d. real standard Gaussian random variables, and z ∈ C is a fixed complex parameter such that |z| ≤ 1. We compute the large N asymptotics for the spectral one-point function E Tr(Y z − w) −1 , with w = E + i0. The energy E is chosen to be comparable with the local eigenvalue spacing of Y z , i.e. we study the small eigenvalues of Y z . The imaginary part of E Tr(Y z − w) −1 is the density of states at the energy E. In particular, we focus on the transitional regime | z| ∼ N −1/2 proving that E Tr(Y z − w) −1 exhibits a one-parameter family of behaviours depending on N 1/2 | z|. Additionally, we prove that E Tr(Y z − w) −1 behaves as in the case of complex Ginibre matrix X for | z| N −1/2 .
In order to study the transitional regime | z| ∼ N −1/2 , we introduce the rescaled variables Section ] it is easy to see that the level spacing of the eigenvalues of Y z close to zero is of order i.e., for |z| < 1 is given by N −2 δ −1 and for |z| = 1 by N −3/2 , which explains the scaling of λ. The unusual N −3/2 scaling in the edge regime |z| = 1 originates from the fact that the density of eigenvalues of the Hermitized matrix features a cubic cusp singularity that has a natural eigenvalue spacing N −3/4 . We now state the main technical result on the large N asymptotics of the one-point function. The main conclusion of the paper will be given as its Corollary . afterwards. Note that the formulas ( ) are considerably simplified when δ = 0, i.e. |z| = 1, in particular, the spectral scaling factor becomes c(N, δ = 0) = N −3/2 .
with Γ any contour around 0 in a counter-clockwise direction, Λ any contour going out from 0 in the direction of e iπ/6 for a while and then going to infinity in the direction e 3πi/5 , and Ω any contour in the fourth quadrant going out from zero in the direction e −iπ/3 and ending in one with an angle e iπ/3 , see Figure . Here ( ) The implicit constant in O(·) depends on C0. Moreover, the integral I (R) (λ, η, δ) is absolutely convergent and is bounded by C(1 + δ) with a constant that depends only on C0 and C1.

F . Depiction of the chosen contours
In Corollary . below we study the behaviour of I (R) (λ, η, δ) in the large | η| regime and we show that, in the large | η| limit, I (R) (λ, η, δ) agrees with the limiting one-point function I (C) (λ, δ) of the complex Ginibre ensemble. We recall from [ , Eq. ( a)] that the limit analogous to ( ) for the complex case is given by The x-integration is over any contour from 0 to e 3iπ/4 ∞, going out from 0 in the direction of the positive real axis, and the y-integration is over any contour around 0 in a counter-clockwise direction.
It is easy to see that along such contours the integral is absolutely convergent. Note that the rhs. of ( ) exactly agrees with [ , Eqs. ( a)-( b)] after the change of variables z * x → x and z * y → y, using the notation therein.

Remark . . From our analysis in Section . (see ( ) later) it actually follows that
of the other two parameters. Remark . . The limiting statement ( ) follows by taking the η limit within the formula ( ), i.e. after the N → ∞ limit is taken. However, we believe that in the regime | η| ≥ C, using a computation similar to the ones in Section . and to the bound [ , Lemmas . -. -. ], but this time on the contours Λ, Ω, one may prove the following stronger result: where the ξ-integration is over any counter-clockwise oriented contour around 0 that does not encircle −1, the a,τ -contours are straight lines, and, using the notation η = z, the functions f and g are defined by ( ) The fact that the integral in ( ) is absolutely convergent follows by the explicit expressions of f and g in ( )-( ). Note that in w = E + i is introduced only to make the a-integration on the imaginary axis absolutely convergent, hence, after the contours deformations described in Section . below, for all the practical purposes we can assume that = 0 and so w = E. Indeed, after deforming the acontour so that it ends in the second quadrant, i.e. in the region {a ∈ C : [a] < 0, [a] > 0}, we can take the limit → 0 + since the integral in ( ) is absolutely convergent for = 0. Note that g(a, 1, η, w) = f (a, w); in particular, we remark that g(a, 1, η, w) is independent of η for any a ∈ C. Furthermore, the function where p i,j,k = p i,j,k (a, τ, ξ) are explicit polynomials in a, τ, ξ which we defer to Appendix B. and δ := 1 − |z| 2 . The indices i, j, k in the definition of p i,j,k denote the N , η and δ power, respectively. We split GN as the sum of G1,N and G2,N since G1,N depends only on |z|, whilst G2,N depends explicitly on η = z, in particular G2,N = 0 if z ∈ R.
. . Choice of the integration contours. From now on we only focus on the regime δ = 0, i.e. |z| = 1. The proof in the case δ ≥ C1 for some large C1 > 0 only requires slightly different choice of contours but otherwise the analysis of the integrals on them is analogous and so we omit the details.
. . . Geometry of { g > 0} in the regime |τ | 1 (see Figure ). In the regime |τ | 1 there is a transition at |1 − τ |η 2 = E 2/3 . In the regime |1 − τ |η 2 E 2/3 there is only one relevant length scale of E −1/3 . On the contrary in the regime |1 − τ |η 2 E 2/3 there are two relevant length scales E −1/3 |1 − τ |η 2 E −1 , the former describing the size of the two connected components of { g > 0} close to 0 and the latter describing the distance to the infinite connected component of { g > 0} in the direction +∞. In Figure we present the level sets of g for various sizes of |1 − τ | and η.
. . . Geometry of { g > 0} in the regime |τ | 1 forη = 0 (see Figure ). In the regime |τ | 1 for η = 0 there is a transitions around |τ | = E. For |τ | E there are two components of { g < 0}, one unbounded at a distance of E −1 to the right of the origin, and a bounded one at a distance of |τ | −1 below the origin. As |τ | approaches E the two components merge but remain separated from the origin at a distance of |τ | −2/3 E −1/3 , see Figure for an illustration.
F . Contour plot of g(·, τ, ηE 1/3 , E) for E > 0 for τ ∈ Ω with |1 − τ | ≤ 1/2. The white lines represent the level set g(·, τ, η, E) = 0, while the black line represents the contour rΛ for the a-integration. All figures are on the same scale E −1/3 , except for the bottom left figure which shows the larger scale E −1/3η2 |1 − τ |, in addition to the E −1/3 length scale of the blue figure eight.
The solid red colours are applied to regions where g > E 2/3 , while the solid blue colours are applied to regions where g < −E 2/3 . . . . Deformation of contours. Now we explain how the contours in ( ) can be deformed. The ξcontour can be freely deformed as long as it does not cross 0 and −1. We can deform the τ -contour as long as [τ ] < 0, then the a-contour has to be deformed accordingly to ensure the absolute convergence of the integral. The a-contour at infinity can be freely deformed, independently of τ , as long as it ends in the second quadrant; on the other hand the way how it goes out from zero depends on τ . Moreover along the deformation of the a-contour we cannot cross the points (−1 ± √ 1 − τ )τ −1 which are the singularities of the term a 2 τ + 2a + 1 in g and GN . In particular, note that the τ and F . Contour plot of g(·, τ, 0, E) for E > 0 for τ ∈ Ω with |τ | 1. The solid white lines represent the level set g(·, τ, 0, E) = 0, while the solid black line represents the contour rΛ for the a-integration. The solid red colours are applied to regions where g > 1, while the solid blue colours are applied to regions where g < −1 F . Contour plot of g(·, τ,ηE 1/3 , E) for 0 ≤ E 1 andη > 0 for τ ∈ Ω with |τ | 1. The solid white lines represent the level set g(·, τ,ηE 1/3 , E) = 0, while the solid black line represents the contour rΛ for the a-integration. The solid red colours are applied to regions where g > 1, while the solid blue colours are applied to regions where g < −1.
a contours cannot be deformed independently: we first deform the τ -contour and then we deform the a-contour accordingly. In the remainder of this section we will always deform the integration contours as described above.
Next, we describe how we concretely deform the integration contours in ( ) in accordance with the rules just described. From now on we denote the ξ-contour by Γ, the τ -contour by Ω, and the a-contour by Λ. In particular, we choose In the following we split the computation of the leading term of ( ) into two parts: (i) in Section . we deal with the regime when either |ξ| ≤ N ω or |aτ | ≤ N 2ω or |τ | ≤ N −ω , for some small fixed ω > 0, (ii) in Section . we deal with the complementary regime when |ξ| and |aτ | are bigger than N ω and |τ | > N −ω .
Lemma . . Let f, g, GN be defined in ( )-( ), and let Γ, Ω, Λ be the contours defined in ( ), then for any large constant C4 > 0, for any E = λN −3/2 , with C −1 4 ≤ λ ≤ C4, and for any η = 0 or e N [f (ξ,E)−g(a,τ,η,E)] aξ 2 τ 1/2 GN (a, τ, ξ, z) Proof. The proof relies on two quantitative lower bounds on g outlined in the following lemmata, the proofs of which we defer to Appendix A. Within these Lemmas and their proofs we deviate from our general convention and the notation f g means that f ≤ cg for a sufficiently small N -independent constant c.
Lemma . . For |τ | ≤ N − we have the following lower bound on g which for clarity we formulate separately depending on the relative sizes of τ, E, a and whether η = 0 or = 0. .
Lemma . . For any 1 ≥ |τ | E with τ ∈ Ω the function We now split the proof of ( ) into three parts, we first prove that the contribution to ( ) in the regime τ ∈ Ω is exponentially small uniformly in ξ ∈ Γ and a ∈ rΛ. Then we prove that the regime a ∈ Λ is also exponentially small for any ξ ∈ Γ and τ ∈ Ω \ Ω. Finally, we conclude that also the contribution for ξ ∈ Γ is negligible.
Using the bound in ( ) we readily conclude that the contribution of the regime τ ∈ Ω is exponentially small and so negligible.
We now consider the regime a ∈ Λ. We split this regime into two cases: (i) |a| ≥ N −10 , (ii) |a| ≤ N −10 . For |a| ≥ N −10 , by ( ) and Lemma . , we readily conclude that In the regime |a| ≤ N −10 we conclude a bound as in ( ) using the explicit form of g in ( ) and that |τ | ≤ 1. This proves that also the regime a ∈ Λ is negligible.
Finally, the fact that the regime ξ ∈ Γ is exponentially small, given that both the regimes τ ∈ Ω and a ∈ Λ are removed, follows exactly as in the proof [ , Lemma . ].
. . Proof of Theorem . . We recall that we only prove the case δ = 0; the case δ ≥ C1 is completely analogous and so omitted. By Lemma . and ( ) we conclude that up to an exponentially small error that we will always ignore in the sequel. In order to compute the leading order of ( ) as N goes to infinity, we use the change of variables where a , ξ are the new integration variables. We get that (omitting the primes, i.e. using the notation a, ξ for the new variables as well to make the notation simpler) da ξ 2 a τ 1/2 e N [f(ξ,w)−g(a,τ,η,w)] G(a, τ, ξ, η) Here we used the asymptotic relations ( ) with f, g, and G defined in ( ). The pre-factor N 3/2 in the leading term of ( ) follows by a simple power counting: a ∼ N 1/2 , ξ ∼ N 1/2 , η ∼ N −1/2 , the volume factor from the Jacobian of the change of variables ( ) gives a factor of N . In order to bound the error term in ( ) we also used the following lemma.
Lemma . . Let f and g be the functions defined in ( ). Then for any fixed α, β, γ ∈ R it holds for some constant C < ∞ which depends only on α, β, γ and on the control parameters C0, C1 from Theorem . .
Proof. The bound in ( ) directly follows from the explicit form of f and g in ( ) and by the fact that on the chosen contours Γ, Ω, rΛ it holds g > 0, f < 0.
Next, using ( ) and Lemma . , we conclude the proof of Corollary . . First of all we notice that the leading term in ( ) does not depend on γ, hence after performing the τ -integration the power of τ that appears in G(a, τ, ξ, η) does not matter. For this reason after the τ -integration we consider G(a, 1, ξ, η), i.e. for convenience we evaluate G at τ = 1. More precisely, by Lemma . it follows that Ω\ Ω 1 τ 1/2 e −g(a,τ, η,λ) G(a, τ, ξ, η) dτ = e −g(a,1, η,λ) G(a, 1, ξ, η) Then, by ( ) together with ( ), it follows that where in the second equality we used the explicit form of G from ( ), and that we can add back the regime a ∈ Λ at the price of a negligible error. This concludes the proof of ( ).
We now present the proof of Lemma . .

A A. A
Proof of Lemma . . The proofs of all lower bounds are similar, hence we will not provide details for all of them. For definiteness we will prove cases .a), .b) and .c) since those already demonstrate the qualitatively different |aτ | 1, |aτ | ∼ 1 and |aτ | 1 regimes.