An inverse visualization for the elliptic law

March 2021

Here are two nice facts on spectra of random matrices of large size nn.

What happens in between lies in the elliptic realm. If AA is a Wigner matrix, the symmetry of AA is indeed equivalent to AA being maximally correlated with its own transpose AA^*, since they are equal. One can thus create intermediary models between Girko and Wigner, by parametrizing the degree of correlation above and below the diagonal: for this, we just take AA to be a fully independent matrix (Girko), and set

Xρ=A+ρA.X_\rho = A + \rho A^*.

The matrix X0=AX_0 = A is a Girko matrix; the matrix X1=A+AX_1 = A + A^* is a symmetric matrix and the matrix X1=AAX_{-1} = A - A^* it is an antisymmetric matrix. In XρX_\rho, the entries above and below the diagonal are correlated, in that if iji \neq j,

Cov((Xρ)i,j,(Xρ)i,j)=2ρVar(Ai,j)=2ρ. \mathrm{Cov}((X_\rho)_{i,j},(X_\rho)_{i,j}) = 2\rho\mathrm{Var}(A_{i,j}) = 2\rho.

It turns out that the eigenvalues of XρX_\rho asymptotically follow the elliptic distribution: they tend to be uniformly distributed inside an ellipse, ie the domain defined by

Real(z)2(1+ρ)2+Imag(z)2(1ρ)21. \frac{\mathrm{Real}(z)^2}{(1 + \rho)^2} + \frac{\mathrm{Imag}(z)^2}{(1 - \rho)^2} \leqslant 1.

The animated picture below is an illustration of this phenomenon, as seen from \infty. elliptic law

For several correlation parameters, I represented the phase portrait of the reciprocical of the characteristic determinant,

qn(z)=det(IzXρ/n) q_n(z)=\det(I - zX_\rho / \sqrt{n})

as a complex function; the white dots are the inverses of the eigenvalues of Xρ/nX_\rho/\sqrt{n}, and the white line is the inverse of the ellipse.

Polynomial convergence

The complex polynomial qnq_n has degree nn. Its (complex) roots are the inverses of the (complex) eigenvalues of A/nA/\sqrt{n}. The central picture in the preceding animation, with ρ=0\rho=0, illustrates a convergence phenomenon regarding qnq_n, which recently appeared in a beautiful paper by Bordenave, Chafaï and García-Zelada.

They showed that if AA is a real Girko matrix entries centered and reduced, then

qnnlawκeF q_n \xrightarrow[n \to \infty]{\mathrm{law}} \kappa \mathrm{e}^{-F}

where κ,F\kappa, F are holomorphic functions; κ\kappa is deterministic,

κ(z)=1z2,\kappa(z) = \sqrt{1 - z^2},

while FF is itself a random function,

F(z)==1Xz F(z) = \sum_{\ell = 1}^\infty X_\ell \frac{z^\ell}{\sqrt{\ell}}

where the XX_\ell are iid standard real Gaussian random variables. The mode of convergence in (5) is the weak convergence of probability measures on the space H(D)\mathbb{H}(\mathbb{D}) – the space of holomorphic functions on the open unit disk, endowed with the classical topology of uniform convergence on compact sets.

Since the limiting random function zκ(z)eF(z)z \mapsto \kappa(z)\mathrm{e}^{-F(z)} does not vanish inside D\mathbb{D}, one can use results like the Hurwitz theorem to show that when nn is large, it is highly unlikely that qnq_n has a root inside D(0,1ϵ)D(0, 1-\epsilon) – thus proving that the eigenvalue of A/nA/\sqrt{n} with highest modulus, say λ1\lambda_1, has lim supλ11+ϵ\limsup |\lambda_1| \leqslant 1 + \epsilon with probability 1o(1)1-o(1).

This is already visible in the picture corresponding to ρ=0\rho=0 above, even though n=100n=100 is pretty small here; the inverse eigenvalues seem to avoid the disk D\mathbb{D}, or be very close to its boundary. Looking at the other pictures, one would merrily suppose that a similar statement holds for every ρ\rho between 1-1 and 11, with no eigenvalue of Xρ/nX_\rho/\sqrt{n} being really far away outside of the ellipse boundary.

Notes and References

The original paper of Girko on the elliptic law... in Russian.

Convergence of the spectral radius of a random matrix through its characteristic polynomial, Bordenave, Chafaï, García-Zelada.

Comments on the circular law, a blogpost by Djalil Chafaï on open problems and recent works around the circular law.

My Julia code for the animated picture.

[1] I will stick to random matrices having real entries, but a similar picture holds with complex entries.