Ginibre Ensemble Research Articles

Point Processes, Hole Events, and Large Deviations: Random Complex Zeros and Coulomb Gases

We consider particle systems (also known as point processes) on the line and in the plane and are particularly interested in “hole” events, when there are no particles in a large disk (or some other domain). We survey the extensive work on hole probabilities and the related large deviation principles (LDP), which has been undertaken mostly in the last two decades. We mainly focus on the recent applications of LDP-inspired techniques to the study of hole probabilities and the determination of the most likely configurations of particles that have large holes. As an application of this approach, we illustrate how one can confirm some of the predictions of Jancovici, Lebowitz, and Manificat for large fluctuation in the number of points for the (two-dimensional) $$\beta $$ -Ginibre ensembles. We also discuss some possible directions for future investigations.

On characteristic polynomials for a generalized chiral random matrix ensemble with a source

We evaluate averages involving characteristic polynomials, inverse characteristic polynomials and ratios of characteristic polynomials for a random matrix taken from a L-deformed chiral Gaussian Unitary Ensemble with an external source Ω. Relation to a recently studied statistics of bi-orthogonal eigenvectors in the complex Ginibre ensemble, see Fyodorov (2017 arXiv:1710.04699), is briefly discussed as a motivation to study asymptotics of these objects in the case of external source proportional to the identity matrix. In particular, for an associated complex bulk/chiral edge scaling regime we retrieve the kernel related to Bessel/Macdonald functions.

Eigenvectors of non normal random matrices

We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e} ^{-n\operatorname{Tr} V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors $\mathbf{v} ,\mathbf{v} '$ associated with distinct eigenvalues $\lambda ,\lambda '$ that are the closest to specified points $z,z'$ in the complex plane, the rescaled inner product \[ \sqrt{n} (\lambda '-\lambda )\langle \mathbf{v} ,\mathbf{v} '\rangle \] is uniformly sub-Gaussian, and give a more precise statement in the case of the Ginibre ensemble.

Powers of Ginibre eigenvalues

We study the images of the complex Ginibre eigenvalues under the power maps $\pi_M: z \mapsto z^M$, for any integer $M$. We establish the following equality in distribution, $$ {\rm{Gin}}(N)^M \stackrel{d}{=} \bigcup_{k=1}^M {\rm{Gin}} (N,M,k), $$ where the so-called Power-Ginibre distributions ${\rm{Gin}}(N,M,k)$ form $M$ independent determinantal point processes. The decomposition can be extended to any radially symmetric normal matrix ensemble, and generalizes Rains' superposition theorem for the CUE and Kostlan's independence of radii to a wider class of point processes. Our proof technique also allows us to recover a result by Edelman and La Croix for the GUE. Concerning the Power-Ginibre blocks, we prove convergence of fluctuations of their smooth linear statistics to independent gaussian variables, coherent with the link between the complex Ginibre Ensemble and the Gaussian Free Field. Finally, some partial results about two-dimensional beta ensembles with radial symmetry and even parameter $\beta$ are discussed, replacing independence by conditional independence.

Extremes of 2d Coulomb gas: universal intermediate deviation regime

In this paper, we study the extreme statistics in the complex Ginibre ensemble of random matrices with complex Gaussian entries, but with no other symmetries. All the N eigenvalues are complex random variables and their joint distribution can be interpreted as a 2d Coulomb gas with a logarithmic repulsion between any pair of particles and in presence of a confining harmonic potential . We study the statistics of the eigenvalue with the largest modulus in the complex plane. The typical and large fluctuations of around its mean had been studied before, and they match smoothly to the right of the mean. However, it remained a puzzle to understand why the large and typical fluctuations to the left of the mean did not match. In this paper, we show that there is indeed an intermediate fluctuation regime that interpolates smoothly between the large and the typical fluctuations to the left of the mean. Moreover, we compute explicitly this ‘intermediate deviation function’ (IDF) and show that it is universal, i.e. independent of the confining potential as long as it is spherically symmetric and increases faster than for large r with an unbounded support. If the confining potential has a finite support, i.e. becomes infinite beyond a finite radius, we show via explicit computation that the corresponding IDF is different. Interestingly, in the borderline case where the confining potential grows very slowly as for with an unbounded support, the intermediate regime disappears and there is a smooth matching between the central part and the left large deviation regime.

Exact mean-energy expansion of Ginibre's gas for coupling constants Γ=2×(oddinteger).

Using the approach of a Vandermonde determinant to the power Γ=Q^{2}/k_{B}T expansion on monomial functions, a way to find the excess energy U_{exc} of the two-dimensional one-component plasma (2DOCP) on hard and soft disks (or a Dyson gas) for odd values of Γ/2 is provided. At Γ=2, the present study not only corroborates the result for the particle-particle energy contribution of the Dyson gas found by Shakirov [Shakirov, Phys. Lett. A 375, 984 (2011)10.1016/j.physleta.2011.01.004] by using an alternative approach, but also provides the exact N-finite expansion of the excess energy of the 2DOCP on the hard disk. The excess energy is fitted to the ansatz of the form U_{exc}=K^{1}N+K^{2}sqrt[N]+K^{3}+K^{4}/N+O(1/N^{2}) to study the finite-size correction, with K^{i} coefficients and N the number of particles. In particular, the bulk term of the excess energy is in agreement with the well known result of Jancovici for the hard disk in the thermodynamic limit [Jancovici, Phys. Rev. Lett. 46, 386 (1981)10.1103/PhysRevLett.46.386]. Finally, an expression is found for the pair correlation function which still keeps a link with the random matrix theory via the kernel in the Ginibre ensemble [Ginibre, J. Math. Phys. 6, 440 (1965)10.1063/1.1704292] for odd values of Γ/2. A comparison between the analytical two-body density function and histograms obtained with Monte Carlo simulations for small systems and Γ=2,6,10,... shows that the approach described in this paper may be used to study analytically the crossover behavior from systems in the fluid phase to small crystals.

Empirical Distributions of Eigenvalues of Product Ensembles

Assume a finite set of complex random variables form a determinantal point process; we obtain a theorem on the limit of the empirical distribution of these random variables. The result is applied to two types of n-by-n random matrices as n goes to infinity. The first one is the product of m i.i.d. (complex) Ginibre ensembles, and the second one is the product of truncations of m independent Haar unitary matrices with sizes $$n_j\times n_j$$ for $$1\le j \le m$$ . Assuming m depends on n, by using the special structures of the eigenvalues we developed, explicit limits of spectral distributions are obtained regardless of the speed of m compared to n. For the product of m Ginibre ensembles, as m is fixed, the limiting distribution is known by various authors, e.g., Gotze and Tikhomirov (On the asymptotic spectrum of products of independent random matrices, 2010. http://arxiv.org/pdf/1012.2710v3.pdf ), Bordenave (Electron Commun Probab 16:104–113, 2011), O’Rourke and Soshnikov (Electron J Probab 16(81):2219–2245, 2011) and O’Rourke et al. (J Stat Phys 160(1):89–119, 2015). Our results hold for any $$m\ge 1$$ which may depend on n. For the product of truncations of Haar-invariant unitary matrices, we show a rich feature of the limiting distribution as $$n_j/n$$ ’s vary. In addition, some general results on arbitrary rotation-invariant determinantal point processes are also derived. In particular, we obtain an inequality for the fourth moment of linear statistics of complex random variables forming a determinantal point process. This inequality is known for the complex Ginibre ensemble only (Hwang in Random matrices and their applications (Brunswick, Maine, 1984), Contemporary Mathematics, American Mathematics Society, Providence, vol 50, pp 145–152, 1986). Our method is the determinantal point process rather than the contour integral by Hwang.

A Central limit theorem for fluctuations in Polyanalytic Ginibre ensembles

We study fluctuations of linear statistics in polyanalytic Ginibre ensembles, a family of point processes describing planar free fermions in a uniform magnetic field at higher Landau levels. Our ma ...

On the distribution of the largest real eigenvalue for the real Ginibre ensemble

Let N−−√+λmaxN+λmax be the largest real eigenvalue of a random N×NN×N matrix with independent N(0,1)N(0,1) entries (the “real Ginibre matrix”). We study the large deviations behaviour of the limiting N→∞N→∞ distribution P[λmax 0t>0, P[λmax<t]=1−14erfc(t)+O(e−2t2). P[λmax<t]=1−14erfc⁡(t)+O(e−2t2). This is a rigorous confirmation of the corresponding result of [Phys. Rev. Lett. 99 (2007) 050603]. We also prove that the left tail is exponential, with correct asymptotics up to O(1)O(1): for t<0t<0, P[λmax<t]=e122π√ζ(32)t+O(1), P[λmax<t]=e122πζ(32)t+O(1), where ζζ is the Riemann zeta-function. Our results have implications for interacting particle systems. The edge scaling limit of the law of real eigenvalues for the real Ginibre ensemble is a rescaling of a fixed time distribution of annihilating Brownian motions (ABMs) with the step initial condition; see [Garrod, Poplavskyi, Tribe and Zaboronski (2015)]. Therefore, the tail behaviour of the distribution of X(max)sXs(max)—the position of the rightmost annihilating particle at fixed time s>0s>0—can be read off from the corresponding answers for λmaxλmax using X(max)s=D4s−−√λmaxXs(max)=D4sλmax.

Asymptotic behavior for log-determinants of several non-Hermitian random matrices

We study the asymptotic behavior for log-determinants of two unitary but non-Hermitian random matrices: the spherical ensembles A-1B; where A and B are independent complex Ginibre ensembles and the truncation of circular unitary ensembles. The corresponding Berry-Esseen bounds and Cramer type moderate deviations are established. Our method is based on the estimates of corresponding cumulants. Numerical simulations are also presented to illustrate the theoretical results.

Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues

Let X be a translation invariant point process on the complex plane and let D be a bounded open set whose boundary has zero Lebesgue measure. We ask what does the point configuration obtained by taking the points of X outside D tell us about the point configuration inside D? We show that for the Ginibre ensemble, it determines the number of points in D. For the translation-invariant zero process of a planar Gaussian Analytic Function, we show that it determines the number as well as the centre of mass of the points in D. Further, in both models we prove that the outside says "nothing more" about the inside, in the sense that the conditional distribution of the inside points, given the outside, is mutually absolutely continuous with respect to the Lebesgue measure on its supporting submanifold.

Hole Probabilities for Finite and Infinite Ginibre Ensembles:

We study the hole probabilities of the infinite Ginibre ensemble ${\mathcal X}_{\infty}$, a determinantal point process on the complex plane with the kernel $\mathbb K(z,w)= \frac{1}{\pi}e^{z\bar w-\frac{1}{2}|z|^2-\frac{1}{2}|w|^2}$ with respect to the Lebesgue measure on the complex plane. Let $U$ be an open subset of open unit disk $\mathbb D$ and ${\mathcal X}_{\infty}(rU)$ denote the number of points of ${\mathcal X}_{\infty}$ that fall in $rU$. Then, under some conditions on $U$, we show that $$ \lim_{r\to \infty}\frac{1}{r^4}\log\mathbb P[\mathcal X_{\infty}(rU)=0]=R_{\emptyset}-R_{U}, $$ where $\emptyset$ is the empty set and $$ R_U:=\inf_{\mu\in \mathcal P(U^c)}\left\{\iint \log{\frac{1}{|z-w|}}d\mu(z)d\mu(w)+\int |z|^2d\mu(z) \right\}, $$ $\mathcal P(U^c)$ is the space of all compactly supported probability measures with support in $U^c$. Using potential theory, we give an explicit formula for $R_U$, the minimum possible energy of a probability measure compactly supported on $U^c$ under logarithmic potential with a quadratic external field. Moreover, we calculate $R_U$ explicitly for some special sets like annulus, cardioid, ellipse, equilateral triangle and half disk.

What is the probability that a large random matrix has no real eigenvalues?

We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2n\times 2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that, $$\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,2k}=\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,0}= -\frac{1}{\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right),$$ where $\zeta$ is the Riemann zeta-function. Moreover, for any sequence of non-negative integers $(k_n)_{n\geq 1}$, $$\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,2k_n}=-\frac{1}{\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right),$$ provided $\lim_{n\rightarrow \infty} \left(n^{-1/2}\log(n)\right) k_{n}=0$.

Continuum percolation for Gaussian zeroes and Ginibre eigenvalues

We study continuum percolation on certain negatively dependent point processes on R-2. Specifically, we study the Ginibre ensemble and the planar Gaussian zero process, which are the two main natural models of translation invariant point processes on the plane exhibiting local repulsion. For the Ginibre ensemble, we establish the uniqueness of infinite cluster in the supercritical phase. For the Gaussian zero process, we establish that a non-trivial critical radius exists, and we prove the uniqueness of infinite cluster in the supercritical regime.

Erratum to: The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

We correct an error in Theorem 12 of the original article concerning the edge scaling limit of the law of real eigenvalues for the real Ginibre ensemble.

Limit theory for geometric statistics of point processes having fast decay of correlations

Let $\mathcal{P}$ be a simple, stationary point process on $\mathbb{R}^{d}$ having fast decay of correlations, that is, its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $\mathcal{P}_{n}:=\mathcal{P}\cap W_{n}$ be its restriction to windows $W_{n}:=[-{\frac{1}{2}}n^{1/d},{\frac{1}{2}}n^{1/d}]^{d}\subset\mathbb{R}^{d}$. We consider the statistic $H_{n}^{\xi}:=\sum_{x\in\mathcal{P}_{n}}\xi(x,\mathcal{P}_{n})$ where $\xi(x,\mathcal{P}_{n})$ denotes a score function representing the interaction of $x$ with respect to $\mathcal{P}_{n}$. When $\xi$ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics and central limit theorems for $H_{n}^{\xi}$ and, more generally, for statistics of the re-scaled, possibly signed, $\xi$-weighted point measures $\mu_{n}^{\xi}:=\sum_{x\in\mathcal{P}_{n}}\xi(x,\mathcal{P}_{n})\delta_{n^{-1/d}x}$, as $W_{n}\uparrow\mathbb{R}^{d}$. This gives the limit theory for nonlinear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model and total edge length of the $k$-nearest neighbors graph) of $\alpha$-determinantal point processes (for $-1/\alpha\in\mathbb{N}$) having fast decreasing kernels, including the $\beta$-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [Ann. Probab. 30 (2002) 171–187] to nonlinear statistics. It also gives the limit theory for geometric $U$-statistics of $\alpha$-permanental point processes (for $1/\alpha\in\mathbb{N}$) as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [Comm. Math. Phys. 310 (2012) 75–98] and Shirai and Takahashi [J. Funct. Anal. 205 (2003) 414–463], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [Stochastic Process. Appl. 56 (1995) 321–335; Statist. Probab. Lett. 36 (1997) 299–306] to show the fast decay of the correlations of $\xi$-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing, and consequently yields the asymptotic normality of $\mu_{n}^{\xi}$ via an extension of the cumulant method.

Dropping the Independence: Singular Values for Products of Two Coupled Random Matrices

We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel-Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.

Test for a universal behavior of Dirac eigenvalues in the complex Langevin method

We apply the complex Langevin (CL) method to a chiral random matrix theory (ChRMT) at non-zero chemical potential and study the nearest neighbor spacing (NNS) distribution of the Dirac eigenvalues. The NNS distribution is extracted using an unfolding procedure for the Dirac eigenvalues obtained in the CL method. For large quark mass, we find that the NNS distribution obeys the Ginibre ensemble as expected. For small quark mass, the NNS distribution follows the Wigner surmise for correct convergence case, while it follows the Ginibre ensemble for wrong convergence case. The Wigner surmise is physically reasonable from the chemical potential independence of the ChRMT. The Ginibre ensemble is known to be favored in a phase quenched QCD at finite chemical potential. Our result suggests a possibility that the originally universal behavior of the NNS distribution is preserved even in the CL method for correct convergence case.

Ornstein–Uhlenbeck diffusion of hermitian and non-hermitian matrices—unexpected links

We compare the Ornstein–Uhlenbeck process for the Gaussian unitary ensemble to its non-hermitian counterpart—for the complex Ginibre ensemble. We exploit the mathematical framework based on the generalized Green’s functions, which involves a new, hidden complex variable, in comparison to the standard treatment of the resolvents. This new variable turns out to be crucial to understand the pattern of the evolution of non-hermitian systems. The new feature is the emergence of the coupling between the flow of eigenvalues and that of left/right eigenvectors. We analyze local and global equilibria for both systems. Finally, we highlight some unexpected links between both ensembles.

Analogies between random matrix ensembles and the one-component plasma in two-dimensions

The eigenvalue PDF for some well known classes of non-Hermitian random matrices — the complex Ginibre ensemble for example — can be interpreted as the Boltzmann factor for one-component plasma systems in two-dimensional domains. We address this theme in a systematic fashion, identifying the plasma system for the Ginibre ensemble of non-Hermitian Gaussian random matrices G, the spherical ensemble of the product of an inverse Ginibre matrix and a Ginibre matrix G1−1G2, and the ensemble formed by truncating unitary matrices, as well as for products of such matrices. We do this when each has either real, complex or real quaternion elements. One consequence of this analogy is that the leading form of the eigenvalue density follows as a corollary. Another is that the eigenvalue correlations must obey sum rules known to characterise the plasma system, and this leads us to an exhibit of an integral identity satisfied by the two-particle correlation for real quaternion matrices in the neighbourhood of the real axis. Further random matrix ensembles investigated from this viewpoint are self dual non-Hermitian matrices, in which a previous study has related to the one-component plasma system in a disk at inverse temperature β=4, and the ensemble formed by the single row and column of quaternion elements from a member of the circular symplectic ensemble.