2022 Seminars

Abstract: We consider quadratic forms evaluated at GOE/GUE eigenvectors, like those studied in the context of quantum unique ergodicity. Under a rank assumption, we show that, in order to compute their extremal statistics, it suffices to replace the eigenvectors with independent Gaussian vectors. By carrying out some representative Gaussian computations, we thus find Gumbel and Weibull limiting distributions for the original problem. Joint work with László Erdős.

Abstract: Embedded random matrix ensembles with $k$-body interactions, usually called EE($k$), introduced 50 years back in the context of nuclear shell model, are now well established to be appropriate for understanding statistical properties of many quantum systems [1]. Say $m$ fermions (or bosons) are in $N$ degenerate single particle states and interacting with $k$-body interactions. Then, with direct product representation of the many-particle states, the $k$ and $m$ fermion space dimensions are $\binom{N}{k}$ and $\binom{N}{m}$ respectively. Now, with a GUE representation for the Hamiltonian ($H$) matrix in the $k$ particle space, the $m$-particle $H$ matrix will be EGUE($k$) - embedded GUE with $k$-body interactions. Similarly, we have EGOE($k$) and EGSE($k$). Note that for $k=m$ we have the classical GOE, GUE and GSE. Recently, using the formulas for the moments up to order 8, it is established that the one-point function, ensemble averaged density of eigenvalues, follows the so called $q$-normal distribution for EGUE($k$) [also for EGOE($k)] with $q$ defined by the fourth moment [2]. The $q$-normal generates Gaussian density for $k << m$ and semi-circle for $k=m$. Unlike the one-point function, till today there is no success in deriving the two-point correlation function for EGUE($k$) or EGOE($k$) even in the limit of $k <<m$. However, recently formulas are obtained for the bivariate moments of the two-point function up to order 8. These moments results, largely obtained by using the underlying $SU(N)$ algebra and the binary correlation approximation [3], will be described in this talk besides first giving a overview of embededd random matrix ensembles in quantum physics.

[1] V.K.B. Kota, Embedded Random Matrix Ensembles in Quantum Physics (Springer, Heidelberg, 2014); V.K.B. Kota and N.D. Chavda, Int. J. Mod. Phys. E {\bf 27}, 1830001 (2018).

[2] Manan Vyas and V.K.B. Kota, J. Stat. Mech.: Theory and Experiment {\bf 2019}, 103103 (2019).

[3] K. K. Mon and J.B. French, Ann. Phys. (N.Y.) {\bf 95}, 90 (1975); L. Benet, T. Rupp, and H.A. Weidenm”{u}ller, Ann. Phys. (N.Y.) {\bf 292}, 67 (2001); V.K.B. Kota, J. Math. Phys. {\bf 46}, 033514 (2005); R.A. Small and S. M”uller, Ann. Phys. (N.Y.) {\bf 356}, 269 (2015); V.K.B. Kota, arXiv:2208.11312 (2022).

Abstract: I will discuss a one dimensional Riesz gas of N particles confined in a harmonic potential. The interaction between any pair of particles at positions x_i and x_j is repulsive and behaves as \sgn(k) |x_i-x_j|^{-k} for i\neq j, where k>-2. For k=-1, this model represents the one dimensional one component plasma, for k\to 0^+, it represents Dyson’s log-gas that appears in random matrix theory and for k=2, it represents the classical Calogero-Moser model. We will first compute the average density in the large N limit explicitly for all k>-2. Next, we will compute the exact average density (large N limit) in the presence of a hard wall at x=w. Finally, I will discuss the statistics of the position of the rightmost particle in the gas, and will compute the explicit large deviation functions of its distribution. We will see that the left tail exhibits a third order phase transition for all k>-2.

Abstract: I will discuss smooth linear statistics of determinantal point processes on the complex plane, and their large scale asymptotics. I will show a CLT in the case where variance stays bounded, and consequently Soshnikov’s theorem is not applicable. The setting is similar to that of Rider and Virág for the complex plane, but replaces analyticity conditions by the assumption that the correlation kernel is reproducing. Our proof is a streamlined version of that of Ameur, Hedenmalm and Makarov for eigenvalues of normal random matrices, where we use the reproducing property in order to compensate for the lack of analyticity and radial symmetries.

Abstract: A partition is a sequence of non-increasing non-negative integers. It has been known that a random distribution of partitions shows similar properties to random matrices. For example, the scaling behavior of the largest increasing subsequence of random permutation described by the Tracy-Widom distribution is one of the primary results in this direction. In this talk, I would discuss random partitions obeying the Schur measure having potentially infinitely many parameters. In particular, I would show that higher analogs of Airy and Pearcey kernels are obtained in the scaling limit of random partitions, and discuss their properties. This talk is based on https://arxiv.org/abs/2012.06424 and https://arxiv.org/abs/2208.07288 in collaboration with A. Zahabi.

Abstract: In a seminal work, Baik/Deift/Johansson established in 1999 that a double scaling limit tailored to the mode of the distribution of the length of longest increasing subsequences in random distributions is given by the beta=2 Tracy-Widom distribution. Since the rate of approximation is rather slow we improve upon this limit by two alternative approaches. First, by a Stirling-type formula we get a numerically accessible approximation of the discrete distribution itself and second, by analytic de-Poissonization (used in this context for the first time), we establish formulas for the first two finite size corrections to the random matrix limit. Both approaches are related to the concept of H-admissible entire functions and the calculations (formula-wise and numerically) are based on representations of generating functions in terms of operator determinants. We derive expansions of the expected value and variance of the length distribution, exhibiting several more terms than previously known.

Abstract: We discuss results on the spectrum of sparse rectangular random matrices with independent entries. Recent results and underlying techniques to derive lower bounds as well as upper bounds for minimal resp. maximal singular values of sparse $n\times N$ random matrices are described in terms of minimal moment assumptions resp. truncated entries and sparsity fractions $p_N$ of order $N P_N \sim (\log N)^{\alpha}$.

Abstract: We will describe the first part of a series of joint works with Luca Lionni and Razvan Gurau in which we generalize the study of HCIZ integrals to higher-order tensors. We obtain developments for the HCIZ integral and its logarithm that are new even in the original setup. Time allowing, we will discuss applications to free probability theory and quantum information theory.

Abstract: Unitary matrix integrals over symmetric functions have a wide variety of applications, including quantum chaos, random processes, enumerative combinatorics, and number theory. In this talk, we derive various novel identities on such integrals, demonstrating universality of the spectral form factor for a broad class of matrix models. We then extend these results and apply them to correlation functions of long-range random walk models, leading to various surprising relations and dualities between them, as well convenient methods for their computation.

Abstract: Topological non-triviality of disordered quantum matter manifests itself in localized states at the boundary of the solid body. The amount of these topological edge states is saved in the topological invariant, whose concrete mathematical nature depends on the symmetries of the system. Generally, for disordered quantum systems, we distinguish between the ten Altland-Zirnbauer symmetry classes, that rely on time reversal invariance, particle-hole conjugation and chiral symmetry. In our work we considered the chiral unitary class AIII and the chiral symplectic class CII in one dimension. Here, the topological invariant is related to the concept of the winding number in complex analysis. We set up a parametric random matrix model for the chiral Hamiltonians, making the Winding number random. Our goal is to obtain the correlation functions of the winding numbers, which we expect to be universal in an unfolding limit. We trace this problem back to an average over ratios of characteristic polynomials, involving the spherical ensemble of matrices K_1^{-1} K_2, where K_1, K_2 are Ginibre distributed. We tackle this problem by employing a technique that exhibits reminiscent supersymmetric structures, while we never carry out any map to superspace.

Abstract: As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of complex random normal matrix models that one finds a complementary error function behavior at the boundary (also called edge) of the droplet as the matrix size increases. Such behavior is seen both in the density, and in the off-diagonal case, where the Faddeeva plasma kernel emerges. These results are neatly expressed with the help of the outward unit normal vector on the edge. We prove that such universal behaviors transcend this class of random normal matrices, being also valid in higher dimensional determinantal point processes, defined on \mathbb C^d. The models under consideration concern higher dimensional generalizations of the determinantal point processes describing the eigenvalues of the complex Ginibre ensemble and the complex elliptic Ginibre ensemble. These models describe a system of particles with mutual repulsion, that are confined to the origin by an external field V(z)=|z|^2 - \tau\Re(z_1^2+…+z_d^2), where 0\leq \tau < 1. Their average density of particles converges to a uniform law on a 2d-dimensional ellipsoidal region. It is on the boundary of this region that we find a complementary error function behavior and the Faddeeva plasma kernel. To the best of my knowledge, this is the first instance of the Faddeeva plasma kernel emerging in a higher dimensional model. Based on arXiv:2208.12676

Abstract: In this talk we will focus on the complex elliptic Ginibre ensemble (eGinUE) and analyze the statistical behavior of its eigenvalues in a suitable scaling limit, known as the weak non-Hermiticity limit. In this limit the asymmetry parameter in the model scales with the matrix dimension and the so obtained 2D limiting point processes generalize the well-known sine and Airy processes from the Gaussian unitary ensemble. Using integrodifferential Painlevé transcendents we will show how the gap functions of the 2D limiting point processes can be evaluated in closed form and how Riemann-Hilbert techniques can subsequently yield precise asymptotic information for the same functions. Based on arXiv:2208.04684 and further ongoing joint work with Alex Little.

Abstract: In this talk we consider a large family of multiplicative statistics of eigenvalues of hermitian random matrix models with a one-cut regular potential. We show that they converge to a universal multiplicative statistics of the Airy2 point process which, in turn, is described in terms of a particular solution to the integro-differential Painlevé II equation. The same solution to this integro-differential equation appeared for the first time in the description of the narrow wedge solution to the KPZ equation, so our results connect the KPZ equation in finite time with random matrix theory in a universal way.

The talk is based on joint work with Promit Ghosal (MIT).

Abstract: The classical paradigm of randomness is the model of independent and identically distributed (i.i.d.) random variables, and venturing beyond i.i.d. is often considered a challenge to be overcome. In this talk, we will explore a different perspective, wherein stochastic systems with constraints in fact aid in understanding fundamental problems. Our constrained systems are well-motivated from statistical physics, including models like the random critical points and determinantal probability measures. These will be used to shed important light on natural questions of relevance in understanding data, including problems of likelihood maximization and dimensionality reduction. En route, we will explore connections to spiked random matrix models and novel asymptotics for the fluctuations of spectrally constrained random systems. Based on the joint works below.

[1] Gaussian determinantal processes: A new model for directionality in data, with P. Rigollet, Proceedings of the National Academy of Sciences, vol. 117, no. 24 (2020), pp. 13207–13213

[2] Fluctuation and Entropy in Spectrally Constrained random fields, with K. Adhikari, J.L. Lebowitz, Communications in Math. Physics, 386, 749–780 (2021).

[3] Maximum Likelihood under constraints: Degeneracies and Random Critical Points, with S. Chaudhuri, U. Gangopadhyay, submitted.

Abstract: Determinants with circular root- and jump-type singularities are of interest in the study of the eigenvalue moduli of random normal matrices.

So far determinants with circular root-type singularities have been unexplored. In the first part of this talk, I will show that such singular determinants have a novel type of asymptotic behavior described in terms of the so-called associated Hermite polynomials.

In the second part, I will focus on determinants with pure jump-type singularities, in the regime where they approach a hard edge. Such determinants give information about the disk counting statistics of coulomb gases near a hard edge and have been unexplored up to now. I will show that the counting statistics in the hard edge regime is considerably wilder than in all previously studied regimes.

The first part of the talk is joint work with S.-S. Byun, and the second part is work in progress with Y. Ameur, J. Cronvall and J. Lenells.

Abstract: I will introduce the notion of operator delocalization, to be contrasted with the more conventional notion of operator growth. I will show that even free quantum mechanical systems, once defined on sufficiently connected networks, can exhibit non-trivial delocalization properties. Some preliminary results, based on a current work in progress, on the connection between operator delocalization and quantum many-body chaos will be discussed.

Abstract: The hard edge Pearcey process is universal in random matrix theory and many other stochastic models. In this talk, we consider gap probability for the thinner/unthinned hard edge Pearcey process over the interval (0, s). By working on the relevant Fredholm determinants, we obtain an integral representation of the gap probability via a Hamiltonian related a system of coupled differential equations and the large gap asymptotics. Moreover, we also establish asymptotic statistical properties of the counting function for the hard edge Pearcey process. Based on joint works with Dan Dai, Shuai-Xia Xu and Luming Yao.

Abstract: A classical problem in probability and combinatorics is to uniformly at random choose a permutation of (1,2, …,N) and find the longest increasing subsequence. Related is the Hammersley process, which essentially takes the permutation length N to be a Poisson distributed random variable with rate z^2, and again looks for the longest increasing subsequence. A remarkable result of Baik, Deift & Johansson tells us that the cumulative distribution function (CDF) of the centred and rescaled length in both the fixed N and Hammersley processes is given by the distribution of the (centred and rescaled) largest eigenvalue of the Laguerre Unitary Ensemble (LUE), in the large N or z limit. Similarly, Borodin & Forrester show that for finite z the Hammersley CDF is given by the distribution of the (centred and rescaled) smallest eigenvalue of the LUE. This allows us to use the “hard-to-soft edge transition” to explore the next order corrections to these CDFs, which expands on and refines results of Baik and Jenkins. We derive expressions for the Hammersley process in terms of Fredholm determinants and Painlevé transcendents, and provide numerical evidence for the corrections to the fixed-N longest increasing subsequence problem.

Abstract: We study the Dyson-Ornstein-Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, relative entropy, and transportation cost. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein-Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions.

Abstract: Exponential functionals of the Brownian motion appear in many different contexts (classical diffusion in random media, quantum scattering, finance,…). I will discuss a recent generalization to the case of matrix Brownian motion. This problem has a natural motivation within the study of quantum scattering on a disordered wire with several conducting channels. I will show that the Wigner-Smith time delay matrix, a fundamental matrix in quantum scattering encoding several characteristic time scales, can be represented as an exponential functional of the matrix BM. I will discuss the relation between this problem of quantum physics and the Dufresne identity, which gives the stationary distribution of such exponential functionals of the BM.

Ref:

Aurélien Grabsch and Christophe Texier. Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity,

J. Phys. A: Math. Theor. 53, 425003 (2020)

Abstract: We use Dumitriu-Edelman tridiagonal matrix models to compute the joint eigenvalue distribution of rank one non-Hermitian perturbations of classical ensembles of random matrices (Gaussian, Wishart, chiral Gaussian), as well as their beta-analogues. Same method can be applied to the computation of the eigenvalue distribution for Hermitian rank one perturbations of chiral Gaussian ensembles, as well as rank one multiplicative non-Hermitian perturbations of each of the above ensembles. Part of the results is a joint work with G. Alpan.

Abstract: In this talk, we will first review some recent results on the eigenvectors of random matrices under fixed-rank deformation, and then we will focus on the limit distribution of the leading eigenvectors of the Gaussian Unitary Ensemble (GUE) with fixed-rank (aka spiked) external source, in the critical regime of the Baik-Ben Arous-Peche (BBP) phase transition. The distribution is given in terms of a determinantal point process with extended Airy kernel. Our result can be regarded as an eigenvector counterpart of the BBP eigenvalue phase transition. The derivation of the distribution makes use of the recently rediscovered eigenvector-eigenvalue identity, together with the determinantal point process representation of the GUE minor process with external source. This is a joint work with Dong Wang (UCAS).

Abstract: The complex elliptic Ginibre ensemble allows one to interpolate between the Ginibre ensemble and the Gaussian Unitary ensemble. It represents a determinantal point process in the complex plane with corresponding kernel, constructed with planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. It provides a unifying approach to rigorously derive several known and new results of local and global spectral statistics. In particular, we prove rigorously some global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support.

We introduce a d-complex dimensional generalization of the elliptic Ginibre ensemble, which interpolates between d-real and d-complex dimensions. In the Hermitian limit, this new ensemble is related to non-interacting Fermions in a trap in d-real dimensions with d-dimensional harmonic oscillator. We provide new local bulk and edge statistics at weak and strong non-Hermiticity for this new ensemble.

This is joint work with Gernot Akemann and Maurice Duits.

Abstract: In this talk, we will discuss the quantitative Tracy-Widom law for the largest eigenvalue of Wigner matrices. More precisely, we will prove that the fluctuations of the largest eigenvalue of a Wigner matrix of size N converge to its Tracy-Widom limit at a rate nearly N^{-1/3}, as N tends to infinity. Our result follows from a quantitative Green function comparison theorem, originally introduced by Erdos, Yau and Yin to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein-Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian ensembles. This is joint work with Kevin Schnelli.

Abstract: In the first part of my talk I propose a model of non-Hermitian normal random matrices for the study of the two-dimensional Coulomb gas at general inverse temperature. We solve this model for N = 2 and discuss a surmise for the nearest neighbour spacing distribution. In the second part I study the statistical behaviour of the symmetry classes of complex symmetric and complex self-dual quaternion matrices. Their nearest neighbour spacing distributions are fitted by 2D Coulomb gases with non-integer values of the inverse temperature 1.4, 2.6 respectively. They have been suggested as the only two symmetry classes with 2D bulk statistics different from the Ginibre ensemble. This is joint work with Gernot Akemann and Adam Mielke.

Abstract: During my talk, I will discuss about two different Riemann-Hilbert approaches to the study of integro-differential Painlevé equations. The study of these equations is motivated by a few interesting physical applications, such as free fermions at finite temperature and stochastic growth models (Kardar-Parisi-Zhang equation). The results I will speak about have been obtained in collaboration with Thomas Bothner, Tom Claeys, Giulio Ruzza and Sofia Tarricone.

Abstract: Schramm-Loewner evolution (SLE) is a stochastic process that describes a random interface appearing in a critical system in two dimensions. Assuming the domain Markov property and the conformal invariance of a random curve, we may argue that the SLE must be driven by a Brownian motion.

Multiple SLE is an extension of SLE that is driven by a many-particle stochastic process and describes multiple random curves.

In contrast to the single curve case, the domain Markov property and the conformal invariance do not fix the stochastic process for multiple SLE due to the nontrivial moduli carried by domains with multiple marked boundary points, leaving the question what the canonical choice of a stochastic process is.

We employed the idea of coupling between SLE and Gaussian free field (GFF) and found that a multiple SLE must be driven by the Dyson model so that it is correctly coupled to GFF.

This talk is based on joint work with Makoto Katori (Chuo University).

Abstract: Random matrix products arise in many problems of statistical physics involving transfer matrices and randomness. A first natural question is to characterize the typical growth, which is controlled by the Lyapunov exponent. It is also interesting to investigate the fluctuations of the random matrix product, which can be characterized through the so-called generalized Lyapunov exponent (GLE) Λ(q), i.e. the cumulant generating function of the logarithm of the norm of the random matrix product. The main question is here to determine explicit formulas. I will review few physical motivations from the physics of Anderson localisation and will focus on the case of matrices from the group SL(2,ℝ), for which the GLE can be obtained by solving a spectral problem. The approach will be illustrated on transfer matrices arising from the study of the Schrödinger equation with a random potential. A possible strategy is to solve the spectral problem perturbatively in q in order to obtain recursively the cumulants: Lyapunov exponent Λ’(0), variance Λ”(0), etc. Finally I will discuss the case of Cauchy disorder, when it is possible to get an exact secular equation for the GLE, from which one can obtain several asymptotically exact results.

Abstract: In this talk, we discuss the fluctuation of f(X) as a matrix, where X is a large square random matrix with centered, independent, identically distributed entries and f is an analytic function. In particular, we show that for a generic deterministic matrix A of the same size as X, the trace of f(X)A is approximately Gaussian which decomposes into two different modes corresponding to tracial and traceless parts of A. We also briefly discuss the proof that mainly relies on resolvents, in particular local laws for products of resolvents. This talk is based on a joint work with László Erdös.

Abstract: I will discuss dictionary learning in the Bayes-optimal setting, in the challenging regime where the matrices to infer have a rank growing linearly with the system size. This is in contrast with most existing literature concerned with the low-rank (i.e., constant-rank) regime. The analysis of this model is possible thanks to a novel combination of the replica method from statistical mechanics together with random matrix theory, coined spectral replica method. It allows us to conjecture variational formulas for the mutual information between hidden representations and the noisy data as well as for the overlaps quantifying the optimal reconstruction error. The proposed method reduces the number of degrees of freedom from O(N^2) matrix entries to O(N) eigenvalues (or singular values), and yields Coulomb gas representations of the mutual information which are reminiscent of matrix models in physics. The main ingredients are the use of Harish-Chandra-Itzykson-Zuber spherical integrals combined with a new replica symmetric decoupling ansatz at the level of the probability distributions of eigenvalues (or singular values) of certain overlap matrices.

Abstract: In the context of loop equations, expansions in 1/N are most familiar. But it’s also possible to perform a scaled high temperature expansion, in which β is proportional to 1/N, and a low temperature expansion in which N is fixed and 1/β is the expansion variable. In fact, there’s a duality between these two expansions. This can be traced back to functional equations for the moments of the corresponding densities. We show that the functional equation can be extended to the mixed moments, and in fact follows from a symmetry of the loop equations for the classical β ensembles.

Abstract: The non-Hermitian Sachdev-Ye-Kitaev model (nHSYK), a model of N strongly-coupled Majorana fermions with random all-to-all q-body non-unitary interactions, is receiving increasing attention because of its possible role as a paradigmatic solvable example of non-Hermitian quantum chaos. In this talk, I will discuss how its local level statistics are well described by (non-Hermitian) random matrix theory (RMT) for q > 2, while for q = 2 it is given by the equivalent of Poisson statistics. For the comparison, we combine exact diagonalization numerical techniques with tools from RMT, in particular complex spacing ratios. Depending on q and N, we identify 19 out of the 38 non-Hermitian universality classes in the nHSYK model, some of which involve universal bulk correlations of classes AI† and AII†, beyond the Ginibre ensembles. At the end, I will also address the probability distribution of the singular values of the nHSYK model, which, in the limit of a large number N of Majoranas, can be related to the weight function of the Al-Salam-Chihara Q-Laguerre polynomials.

Abstract: We consider a multiplicative perturbation of the form $UA(t)$ where $U$ is a unitary random matrix and $A = diag (t,1,…,1)$. This so-called “$UA$ model” was first introduced by Fyodorov in 2001 for its applications in scattering theory. In this talk, I will give a general description of the eigenvalue trajectories obtained by varying the parameter $t$ and discuss a flow of deterministic domains that allows separating the outlier resulting from the rank-one perturbation from the typical eigenvalues for all sub-critical timescales. The results are obtained under generic assumptions on $U$ that hold for a variety of unitary random matrix models.

Abstract: Left and right eigenvectors of non-Hermitian random matrices can be chosen so as to form a biorthogonal family. One of the most relevant statistics about them is the “matrix of overlaps”, introduced in the late ’90s by Chalker & Mehlig and studied since in different models, using a variety of techniques. I will present some recent progress on the study of overlaps between eigenvectors in the Truncated Unitary Ensembles (truncations of Haar-distributed unitary matrices) and related models of random matrices.

Abstract: In this talk we investigate the Wasserstein distance between the empirical spectral distribution of non-Hermitian random matrices and the Circular Law. For general entry distributions, I present a nearly optimal rate of convergence in 1-Wasserstein distance of order n^(-1/2+ε) and show that the optimal rate n^(-1/2) is attained by Ginibre matrices. This reveals that the expected transport cost of complex eigenvalues to the uniform measure on the unit disk decays faster compared to that of i.i.d. points, which is known to include a logarithmic factor. We shall also discuss the results from a point of view of random geometry, which will be accompanied by illustrative simulations.