Dysonian dynamics of the Ginibre ensemble

Dynamical fluctuations in the Riesz gas

We consider an infinite system of particles on a line performing identical Brownian motions and interacting through the |x-y|^{-s} Riesz potential, causing the overdamped motion of particles. We investigate fluctuations of the integrated current and the position of a tagged particle. We show that for 0<s<1, the standard deviations of both quantities grow as t^{s/2(1+s)}. When s>1, the interactions are effectively short-ranged, and the universal subdiffusive t^{1/4} growth emerges with only amplitude depending on the exponent s. We also show that the two-time correlations of the tagged-particle position have the same form as for fractional Brownian motion.

The Brown measure of the free multiplicative Brownian motion

The free multiplicative Brownian motion $$b_{t}$$ b t is the large-N limit of the Brownian motion on $$\mathsf {GL}(N;\mathbb {C}),$$ GL ( N ; C ) , in the sense of $$*$$ ∗ -distributions. The natural candidate for the large-N limit of the empirical distribution of eigenvalues is thus the Brown measure of $$b_{t}$$ b t . In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $$\Sigma _{t}$$ Σ t that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $$W_{t}$$ W t on $$\overline{\Sigma }_{t},$$ Σ ¯ t , which is strictly positive and real analytic on $$\Sigma _{t}$$ Σ t . This density has a simple form in polar coordinates: $$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ W t ( r , θ ) = 1 r 2 w t ( θ ) , where $$w_{t}$$ w t is an analytic function determined by the geometry of the region $$\Sigma _{t}$$ Σ t . We show also that the spectral measure of free unitary Brownian motion $$u_{t}$$ u t is a “shadow” of the Brown measure of $$b_{t}$$ b t , precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

Real spectra of large real asymmetric random matrices

When a randomness is introduced at the level of real matrix elements, depending on its particular realization, a pair of eigenvalues can appear as real or form a complex conjugate pair. We show that in the limit of large matrix size the density of such real eigenvalues is proportional to the square root of the asymptotic density of complex eigenvalues continuated to the real line. This relation allows one to calculate the real densities up to a normalization constant, which is then applied to various examples, including heavy-tailed ensembles and adjacency matrices of sparse random regular graphs.

Eikonal formulation of large dynamical random matrix models

The standard approach to dynamical random matrix models relies on the description of trajectories of eigenvalues. Using the analogy from optics, based on the duality between the Fermat principle (rays) and the Huygens principle (wavefronts), we formulate the Hamilton-Jacobi dynamics for large random matrix models. The resulting equations describe a broad class of random matrix models in a unified way, including normal (Hermitian or unitary) as well as strictly non-normal dynamics. This formalism applied to Brownian bridge dynamics allows one to calculate the asymptotics of the Harish-Chandra-Itzykson-Zuber integrals.

Universality of local spectral statistics of products of random matrices

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→∞. The singular value statistics is described by a determinantal point process with a kernel that interpolates between Gaussian unitary ensemble statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit N→∞, 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 a. We find that in addition to the standard scaling regimes, there is a transitional regime which interpolates between the hard edge and the bulk. We conjecture that these results are universal, and that they apply to a broad class of products of random matrices from the Gaussian basin of attraction, including correlated matrices. We corroborate this conjecture by numerical simulations. Additionally, we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of Dyson Brownian motion with the initial condition given by equidistant positions, with the crucial difference that this equivalence holds only locally. Finally, we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum.

Necklace-State-Mediated Anomalous Enhancement of Transport in Anderson-Localized non-Hermitian Hybrid Systems

Non-Hermiticity is known to manifest interesting modifications in the transport properties of complex systems. We report an intriguing regime of transport of hybrid quasiparticles in a non-Hermitian setting. We calculate the probability of transport, quantified by the Thouless conductance, of hybrid plasmons under varying degrees of disorder. With increasing disorder, we initially observe an expected decrease in average transmission, followed by an anomalous rise at localizing disorder. The behavior originates from the confluence of hybridization and non-Hermiticity, in which the former realizes the aggregation of eigenvalues migrating under disorder, while the latter enables energy transfer between the eigenmodes. We find that the enhanced transmission is mediated by quasiparticle hopping over various Anderson-localized states within the so-formed necklace states. We note that, in this scenario, all configurations exhibit the formation of necklace states and enhanced transport, unlike the conventionally known behavior of necklace states which only occurs in rare configurations.

From Synaptic Interactions to Collective Dynamics in Random Neuronal Networks Models: Critical Role of Eigenvectors and Transient Behavior

The study of neuronal interactions is at the center of several big collaborative neuroscience projects (including the Human Connectome Project, the Blue Brain Project, and the Brainome) that attempt to obtain a detailed map of the entire brain. Under certain constraints, mathematical theory can advance predictions of the expected neural dynamics based solely on the statistical properties of the synaptic interaction matrix. This work explores the application of free random variables to the study of large synaptic interaction matrices. Besides recovering in a straightforward way known results on eigenspectra in types of models of neural networks proposed by Rajan and Abbott (2006), we extend them to heavy-tailed distributions of interactions. More important, we analytically derive the behavior of eigenvector overlaps, which determine the stability of the spectra. We observe that on imposing the neuronal excitation/inhibition balance, despite the eigenvalues remaining unchanged, their stability dramatically decreases due to the strong nonorthogonality of associated eigenvectors. This leads us to the conclusion that understanding the temporal evolution of asymmetric neural networks requires considering the entangled dynamics of both eigenvectors and eigenvalues, which might bear consequences for learning and memory processes in these models. Considering the success of free random variables theory in a wide variety of disciplines, we hope that the results presented here foster the additional application of these ideas in the area of brain sciences.

What drives transient behavior in complex systems?

We study transient behavior in the dynamics of complex systems described by a set of nonlinear ordinary differential equations. Destabilizing nature of transient trajectories is discussed and its connection with the eigenvalue-based linearization procedure. The complexity is realized as a random matrix drawn from a modified May-Wigner model. Based on the initial response of the system, we identify a novel stable-transient regime. We calculate exact abundances of typical and extreme transient trajectories finding both Gaussian and Tracy-Widom distributions known in extreme value statistics. We identify degrees of freedom driving transient behavior as connected to the eigenvectors and encoded in a nonorthogonality matrix T_{0}. We accordingly extend the May-Wigner model to contain a phase with typical transient trajectories present. An exact norm of the trajectory is obtained in the vanishing T_{0} limit where it describes a normal matrix.

Eigenvector statistics of the product of Ginibre matrices

We develop a method to calculate left-right eigenvector correlations of the product of m independent N×N complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of examples for small m and N. We conjecture that the integrated overlap between left and right eigenvectors is given by the formula O=1+(m/2)(N-1) and support this conjecture by analytical and numerical calculations. We derive an analytical expression for the limiting correlation density as N→∞ for the product of Ginibre matrices as well as for the product of elliptic matrices. In the latter case, we find that the correlation function is independent of the eccentricities of the elliptic laws.

Quaternionic R transform and non-Hermitian random matrices

Using the Cayley-Dickson construction we rephrase and review the non-Hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp, and I. Zahed, Nucl. Phys. B 501, 603 (1997)], that generalizes the free probability calculus to asymptotically large non-Hermitian random matrices. The main object in this generalization is a quaternionic extension of the R transform which is a generating function for planar (noncrossing) cumulants. We demonstrate that the quaternionic R transform generates all connected averages of all distinct powers of X and its Hermitian conjugate X†: 〈〈1/NTrX(a)X(†b)X(c)...〉〉 for N→∞. We show that the R transform for Gaussian elliptic laws is given by a simple linear quaternionic map R(z+wj)=x+σ(2)(μe(2iϕ)z+wj) where (z,w) is the Cayley-Dickson pair of complex numbers forming a quaternion q=(z,w)≡z+wj. This map has five real parameters Rex, Imx, ϕ, σ, and μ. We use the R transform to calculate the limiting eigenvalue densities of several products of Gaussian random matrices.