We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant $z\in\mathbb{C}$. We prove an optimal lower tail estimate on this singular value in the critical regime where $z$ is around the spectral edge thus improving the classical bound of [Sankar, Spielman, Teng, 2006] in the edge regime. Lacking Brézin-Hikami formulas in the real case, we rely on the superbosonization formula [Littelmann, Sommers, Zirnbauer, 2008].
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 s…
We derive an accurate lower tail estimate on the lowest singular value $σ_1(X-z)$ of a real Gaussian (Ginibre) random matrix $X$ shifted by a complex parameter $z$. Such shift effectively changes the upper tail behaviour of the condition number $κ(X-z)$ from the slower $\mathbf{P}(κ(X-z)\ge t)\lesssim 1/t$ decay typical for real Ginibre matrices to the faster $1/t^2$ decay seen for complex Ginibre matrices as long as $z$ is away from the real axis. This sharpens and resolves a recent conjecture in [arXiv:2005.08930] on the regularizing effect of the real Ginibre ensemble with a genuinely compl…
We study the least singular value of the $n\times n$ matrix $H-z$ with $H=A_0+H_0$, where $H_0$ is drawn from the complex Ginibre ensemble of matrices with iid Gaussian entries, and $A_0$ is some general $n\times n$ matrix with complex entries (it can be random and in this case it is independent of $H_0$). Assuming some rather general assumptions on $A_0$, we prove an optimal tail estimate on the least singular value in the regime where $z$ is around the spectral edge of $H$ thus generalize the recent result of Cipolloni, Erdős, Schröder arxiv:1908.01653 to the case $A_0\ne 0$. The result impr…
The complex elliptic Ginibre ensemble with coupling $τ$ is a complex Gaussian matrix interpolating between the Gaussian Unitary Ensemble (GUE) and the Ginibre ensemble. It has been known for some time that its eigenvalues form a determinantal point process in the complex plane. A recent result of Kanazawa and Kieburg (arXiv:1804.03985) shows that the singular values form a Pfaffian point process. In this paper we turn to consider an extended elliptic Ginibre ensemble, which connects the GUE and the spiked Wishart matrix, and prove that the singular values still build a Pfaffian point process w…
It has been shown by Akemann, Ipsen and Kieburg that the squared singular values of products of $M$ rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that admits a representation in terms of Meijer G-functions. We prove the universality of the local statistics of the squared singular values, namely, the bulk universality given by the sine kernel and the edge universality given by the Airy kernel. The proof is based on the asymptotic analysis for the double contour integral representation of…
Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin …
Let $\sqrt{N}+λ_{max}$ be the largest real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix'). We study the large deviations behaviour of the limiting $N\rightarrow \infty$ distribution $P[λ_{max}<t]$ of the shifted maximal real eigenvalue $λ_{max}$. In particular, we prove that the right tail of this distribution is Gaussian: for $t>0$, \[ P[λ_{max}<t]=1-\frac{1}{4}\mbox{erfc}(t)+O\left(e^{-2t^2}\right). \] This is a rigorous confirmation of the corresponding result of Forrester and Nagao. We also prove that the left tail is exponential: fo…
Akemann, Ipsen, and Kieburg showed recently that the squared singular values of a product of M complex Ginibre matrices are distributed according to a determinantal point process. We introduce the notion of a polynomial ensemble and show how their result can be interpreted as a transformation of polynomial ensembles. We also show that the squared singular values of the product of M-1 complex Ginibre matrices with one truncated unitary matrix is a polynomial ensemble, and we derive a double integral representation for the correlation kernel associated with this ensemble. We use this to calculat…
The singular values squared of the random matrix product $Y = G_r G_{r-1} \cdots G_1 (G_0 + A)$, where each $G_j$ is a rectangular standard complex Gaussian matrix while $A$ is non-random, are shown to be a determinantal point process with correlation kernel given by a double contour integral. When all but finitely many eigenvalues of $A^*A$ are equal to $bN$, the kernel is shown to admit a well-defined hard edge scaling, in which case a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of $0<b<1$ is in…
We consider the squared singular values of the product of $M$ standard complex Gaussian matrices. Since the squared singular values form a determinantal point process with a particular Meijer G-function kernel, the gap probabilities are given by a Fredholm determinant based on this kernel. It was shown by Strahov \cite{St14} that a hard edge scaling limit of the gap probabilities is described by Hamiltonian differential equations which can be formulated as an isomonodromic deformation system similar to the theory of the Kyoto school. We generalize this result to the case of finite matrices by …
We consider various asymptotic scaling limits $N\to\infty$ for the $2N$ complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding kernel for different potentials. The first part is devoted to the symplectic Ginibre ensemble with the Gaussian potential. We obtain the asymptotic at the edge of the spectrum in the vicinity of the real line. The unifying form of the kernel allows us to make contact with the bulk scaling al…
Let $X$ be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of $GX$ and $TX$, where $G$ is a complex Ginibre matrix and $T$ is a truncated unitary matrix. We also consider the product of $X$ and several complex Ginibre/truncated unitary matrices. As an application, we derive the precise condition for the squared singular values of the product of several truncated unitary matrices to follow a polynomial ensemble. We also consider the sum $H + M$ where $H$ is a GUE …
We consider words $G_{i_1} \cdots G_{i_m}$ involving i.i.d. complex Ginibre matrices, and study tracial expressions of their eigenvalues and singular values. We show that the limit distribution of the squared singular values of every word of length $m$ is a Fuss-Catalan distribution with parameter $m+1$. This generalizes previous results concerning powers of a complex Ginibre matrix, and products of independent Ginibre matrices. In addition, we find other combinatorial parameters of the word that determine the second-order limits of the spectral statistics. For instance, the so-called coperiod…
We prove that the squared singular values of a fixed matrix multiplied with a truncation of a Haar distributed unitary matrix are distributed by a polynomial ensemble. This result is applied to a multiplication of a truncated unitary matrix with a random matrix. We show that the structure of polynomial ensembles and of certain Pfaffian ensembles is preserved. Furthermore we derive the joint singular value density of a product of truncated unitary matrices and its corresponding correlation kernel which can be written as a double contour integral. This leads to hard edge scaling limits that also…
Let $λ_{max}$ be a shifted maximal real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix') in the $N\to\infty$ limit. It was shown by Poplavskyi, Tribe, Zaboronski \cite{PZT} that the limiting distribution of the maximal real eigenvalue has $s\to-\infty$ asymptotics $$\mathbb{P} [ λ_{max} < s ] = {\rm e}^{\frac{1}{2\sqrt{2π}} ζ(\frac32)s + \mathcal{O}(1)},$$ where $ζ$ is the Riemann zeta-function. This limiting distribution was expressed by Baik, Bothner \cite{BB18} in terms of the solution $q(x)$ of a certain Zakharov-Shabat inverse scatter…
We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed unitary matrices with appropriate dimensional growth. In the general case we develop a new approach to obtain complex integral representations for densities of measures whose Stieltjes transforms satisfy algebraic equations of a certain type. In the special cases in which at most one factor of the product is a complex Gaussian we derive elementary expression…
Let $σ_n(\cdot)$ denote the least singular value of a $n \times n$ matrix. It is well-known that $\mathbb{P}[σ_n(A) \le \varepsilon] \le \varepsilon n$ if $A$ is drawn from the real Ginibre ensemble of $n \times n$ matrices and $\mathbb{P}[σ_n(A) \le \varepsilon] \le \varepsilon^2 n^2$ if $A$ is drawn from the complex Ginibre ensemble. In this paper, we will show a similar phenomenon occurs for sparse random matrices.
A rank-two spherical transform for eigenvector and singular-vector overlaps
Institut Louis Bachelier（路易·巴舍利耶研究所）；Econophysics Lab, Institut Louis Bachelier（经济物理学实验室，路易·巴舍利耶研究所）；LadHyX, UMR CNRS 7646, Ecole Polytechnique, Institut Polytechnique de Paris（拉迪赫实验室，法国国家科学研究中心联合研究单位7646，巴黎综合理工学院，巴黎理工学院）
Let $G_n$ be an $n \times n$ matrix with real i.i.d. $N(0,1/n)$ entries, let $A$ be a real $n \times n$ matrix with $\Vert A \Vert \le 1$, and let $γ\in (0,1)$. We show that with probability $0.99$, $A + γG_n$ has all of its eigenvalue condition numbers bounded by $O\left(n^{5/2}/γ^{3/2}\right)$ and eigenvector condition number bounded by $O\left(n^3 /γ^{3/2}\right)$. Furthermore, we show that for any $s > 0$, the probability that $A + γG_n$ has two eigenvalues within distance at most $s$ of each other is $O\left(n^4 s^{1/3}/γ^{5/2}\right).$ In fact, we show the above statements hold in the mo…
Let $R_n$ be a $n \times n$ random matrix with i.i.d. subgaussian entries. Let $M$ be a $n \times n$ deterministic matrix with norm $\lVert M \rVert \le n^γ$ where $1/2<γ<1$. The goal of this paper is to give a general estimate of the smallest singular value of the sum $R_n + M$, which improves an earlier result of Tao and Vu.
The singular values of a product of $M$ independent Ginibre matrices of size $N\times N$ form a determinantal point process. Near the soft edge, as both $M$ and $N$ go to infinity in such a way that $M/N\to α$, $α>0$, a scaling limit emerges. We consider a gap probability for the corresponding limiting determinantal process, namely, the probability that there are no particles in the interval $(a,+\infty)$. We derive a Tracy-Widom-like formula in terms of the unique solution of a certain matrix Riemann-Hilbert problem of size $2 \times 2$. The right-tail asymptotics for this solution is obtaine…
We derive a lower bound on the smallest singular value of a random $d$-regular matrix, that is, the adjacency matrix of a random $d$-regular directed graph. More precisely, let $C_1<d< c_1 n/\log^2 n$ and let $\mathcal{M}_{n,d}$ be the set of all $0/1$-valued square $n\times n$ matrices such that each row and each column of a matrix $M\in \mathcal{M}_{n,d}$ has exactly $d$ ones. Let $M$ be uniformly distributed on $\mathcal{M}_{n,d}$. Then the smallest singular value $s_{n} (M)$ of $M$ is greater than $c_2 n^{-6}$ with probability at least $1-C_2\log^2 d/\sqrt{d}$, where $c_1$, $c_2$, $C_1$, a…
Very recently, we have shown how the harmonic analysis approach can be modified to deal with products of general Hermitian and complex random matrices at finite matrix dimension. In the present work, we consider the particular product of a multiplicative Pólya ensemble on the complex square matrices and a Gaussian unitary ensemble (GUE) shifted by a constant multiplicative of the identity. The shift shall show that the limiting hard edge statistics of the product matrix is weakly dependent on the local spectral statistics of the GUE, but depends more on the global statistics via its Stieltjes …
Some properties that nominally involve the eigenvalues of Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This independence is remarkable given the well known phenomenon of eigenvalue repulsion. The structure of this decomposition reveals that several existing observations about large $n$ limits of the GUE are in fact manifestations of phenom…
We analyse the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by M. Meckes for the Abelian case. We show that for regular sequences of groups the limiting distribution of eigenvalues (resp. singular values) is a mixture of eigenvalue (resp. singular value) distributions of Ginibre matrices with the directing measure being related to the limiting behavior of the Plancherel measure of the sequence of groups. In particular for the sequence of symmetric groups, the limiting distri…
We derive exact analytical expressions for correlation functions of singular values of the product of $M$ Ginibre matrices of size $N$ in the double scaling limit $M,N\rightarrow \infty$. The singular value statistics is described by a determinantal point process with a kernel that interpolates between GUE statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit, $N\rightarrow \infty$, the interpolation parameter is given by the limiting quotient $a=N/M$. One of our goals is to find an explicit form of the kernel at the hard edge, in the bulk and at the soft edge for any …
