Let $(X_{jk})_{j,k\geq1}$ be an infinite table of complex random variables and set $X:=(X_{j,k})_{1\leq j,k\leq n}$. If $X_{11}$ is Gaussian then $X$ belongs to the so called Ginibre Ensemble. Consider the random unitary matrices $U$ and $V$ such that $X=UDV$ where $D=\mathrm{diag}(s_1,\ldots,s_n)$ and where $s_1,\ldots,s_n$ are the singular values of $X$, i.e. the eigenvalues of $\sqrt{XX^*}$. When $X_{11}$ is Gaussian, the law of $X$ is rotationally invariant, and the matrices $U$ and $V$ are distributed according to the Haar law on the unitary group $\mathbb{U}_n$. The Gaussian version of the Marchenko-Pastur theorem tells us that with probability one, the counting probability distribution of the singular values, appropriately scaled, tends weakly to the quartercircular  law as $n\to\infty$.

The Marchenko-Pastur theorem is universal in the sense that it holds with the same limit beyond the Gaussian case provided that $X_{11}$ has moments identical to the Gaussian up to the order 2. One can ask if a similar statement holds for the eigenvectors, i.e. for the matrices $U$ and $V$. Are they asymptotically Haar distributed? For instance, one may ask if $W_2(\mathcal{L}(U),\mathrm{Haar}(\mathbb{U}_n))$ tends to zero as $n\to\infty$, where $W_2$ is the Wasserstein coupling distance. The distance choice is important. One may consider  many other distances including for instance the Fourier distance $\sup_g|\Phi_\mu(g)-\Phi_\nu(g)|$ where $\Phi_\mu$ denotes the Fourier transforrm of $\mu$ (characteristic function). A weakened version of this statement consist in asking if linear functionals of $U$ and $V$ behave asymptotically as Brownian bridges. Indeed, it is well known that linear functionals of the Haar law on the unitary group behave asmptotically like this. Silverstein has done some work in this direction. Of course, one can ask the same question for the eigenvectors in the Girko circular law and in the Wigner theorem. One can guess that a finite fourth moment assumption on $X_{11}$ is needed, otherwise the top of the spectrum will blow up and the corresponding eigenvectors will maybe localize.

If you do not trust me, just do simulations or… computations! There is here potentially a whole line of research, sparsely explored for the moment. If you like free probability, you may ask if $U’XV’$ is close to $X$ when $U’$ and $V’$ are Haar distributed and independent of $X$.

There is some literature on the behavior of eigenvectors of deterministic matrices under perturbations of the entries of the matrix, see e.g. the book of Bhatia (ch. VII). Among many results, if $A$ and $B$ are two invertible $n\times n$ complex matrices with respective polar unitary factors $U_A$ and $U_B$ in their polar factorization then for any unitary invariant norm $\left\Vert\cdot\right\Vert$ we have

$\displaystyle\left\Vert U_A-U_B\right\Vert\leq 2\frac{\left\Vert A-B\right\Vert}{\left\Vert A^{-1}\right\Vert^{-1}+\left\Vert B^{-1}\right\Vert^{-1}}.$

The eigenvectors are more sensitive than the bulk of the spectrum to perturbations on $X$, and one may understand this by remembering that for a normal matrix, they are arg-suprema while the eigenvalues are suprema. Also, one can guess that the asymptotic uniformization of the eigenvectors may be even sensitive to the skewness of the law of $X_{11}$.

It is well known that the $k$-dimensional projection of the uniform law on the sphere of $\mathbb{R}^n$ of radius $\sqrt{n}$ tends to the Gaussian law as $n\to\infty$. By viewing $\mathbb{U}_n$ as a bunch of exchangeable spheres, one can guess that the Haar law on the unitary group, appropriately scaled, will converge in some sense to the Brownian sheet bridge as the dimension tends to infinity. Recent addition to this post: this was proved in a paper by Donati-Martin and Rouault! We conjecture that this result is universal for  the eigenvectors matrix of random matrices with i.i.d. entries and moments identical to the Gaussian moments up to order $4$.

The uniformization of the eigenvectors of random matrices is related to their delocalization, a phenomenon recently investigated by Erdös, Schlein, Ramirez, Yau, Tao, Vu, as a byproduct of their analysis of the universality of local statistics of the spectrum. This is a huge contrast with the well known Anderson localization phenomenon in mathematical physics for random Schrödinger operators.

The unitary group $\mathbb{U}_n$ is a purely $\ell^2$ object. Its $\ell^1$ analogue is the Birkhoff polytope of doubly stochastic matrices, also known as the transportation polytope,  assignment polytope, or perfect matching polytope, but this is another story…

This post benefined from discussions with Charles Bordenave and Florent Benaych-Georges.

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