Universality of local spectral statistics of products of random matrices

Gernot Akemann, Zdzislaw Burda, and Mario Kieburg

Phys. Rev. E 102, 052134 – Published 30 November 2020

Abstract

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 $M, N \to \infty$ . 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 $N \to \infty$ , 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 $a$ . 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.

Received 7 September 2020
Accepted 8 November 2020

DOI:https://doi.org/10.1103/PhysRevE.102.052134

Physics Subject Headings (PhySH)

Brownian motion Critical phenomena

Integrable systems Random matrix theory

Statistical Physics & Thermodynamics

Authors & Affiliations

Gernot Akemann ^*

Faculty of Physics, Bielefeld University, Postfach 100131, D-33501 Bielefeld, Germany

Zdzislaw Burda ^†

Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, PL-30059 Krakow, Poland

Mario Kieburg ^‡

School of Mathematics and Statistics, University of Melbourne, 813 Swanston Street, Parkville, Melbourne VIC 3010, Australia

^*akemann@physik.uni-bielefeld.de
^†zdzislaw.burda@agh.edu.pl
^‡m.kieburg@unimelb.edu.au

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 102, Iss. 5 — November 2020

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
The macroscopic level density (49) obtained in the limit $N \to \infty$ for various values of a fixed number of matrices $M$ multiplied. For increasing $M$ the spectrum concentrates at $\tilde{ϕ} = π - ϕ = 0$ which constitutes the deep bulk of the spectrum. Yet, the tail supported inside the interval $] 0, π [$ still comprises roughly $N / M$ eigenvalues which is finite when $N = O (M)$ and even creates a mesoscopic, soft bulk of eigenvalues when $N ≫ M$ . We have reflected the angle so that the hard edge lies at the origin which agrees with the picture of the hard edge of the product matrix $Y$ .
Reuse & Permissions
Figure 2
The properly unfolded level density (118) at the soft edge for various ratios $a = N / M$ . The bulk of eigenvalues is to the left. The peak of the distributions at the origin is the price we have to pay for unfolding, which generates a coordinate singularity since the mesoscopic level density vanishes there. Note that the individual eigenvalues are shifted by $\frac{1}{2}$ and do not lie on the integers.
Reuse & Permissions
Figure 3
The total eigenvalue counts (histogram values are proportional to the level density) of the six ensembles. The bin size is equal to 0.1 and has been chosen to be significantly smaller than the peaks' widths in order to see local fluctuations of the level density. Each ensemble consists of $10^{4}$ configurations of a product of $M = 500$ matrices of size $60 \times 60$ . Therefore, each histogram comprises $6 \times 10^{5}$ eigenvalues. We have slightly rescaled the eigenvalues to be able to apply the same bin size.
Reuse & Permissions
Figure 4
The unfolded microscopic level density of the six ensembles 1–6 of product matrices at the soft edge (symbols) compared to the analytical result (118) in combination with (109) (smooth curves), with a fitted effective value $a$ that replaces the ratio $N / M$ . Even for the Ginibre case (upper left plot), for which we have analytically studied the double scaling limits, we cannot rely completely on the relation $a = N / M$ , which would be $a = 60 / 500 = 0.12$ (blue dashed curve) in the present case. Due to the finite matrix size, some deviations show up for smaller eigenvalues. Therefore, we have additionally plotted the result for $a = 50 / 500 = 0.1$ (brown curve) which fits the 10th largest eigenvalue best. Certainly, similar deviations for the other ensembles can be explained by finite size effects, and should be always taken into account when comparing with empirical data. The matrix size is $N = 60$ , the number of matrices multiplied is $M = 500$ , and the ensemble size is $10^{4}$ for all six ensembles.
Reuse & Permissions
Figure 5
The real (left plot) and imaginary (right plot) parts of the digamma function (B2) in the upper right quadrant of the complex plane. Note that we have mapped the whole quarter plane to a square by the monotonicity preserving map $(k, t) \mapsto (k / [1 + k], t / [1 + t])$ . The black curve is defined by $Im [ψ (k + i t)] = π / 2$ and the dashed black line at $k = \frac{1}{2}$ indicates when the $Im [ψ (k + i t)] < 0$ is satisfied for all positive $t > 0$ . The height of the real and imaginary parts is color coded in a density plot where purple and red are either $- \infty$ and $+ \infty$ for $Re [ψ (k + i t)]$ or 0 and $π / 2$ for $Im [ψ (k + i t)]$ . The vectors highlight the local steepest ascent of the function, in particular, they show our claim that $Im [ψ (k + i t)]$ is increasing in $t$ and $Re [ψ (k + i t)]$ is increasing in $k$ in this regime.
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics