Self-similarity in the circular unitary ensemble

Self-similarity in the circular unitary ensemble, Discrete Analysis, 2016:9, 15 pp.

The distribution of the eigenvalues of random matrices is a central topic of research. It includes famous results such as Wigner’s semicircle law from 1955, which states that the distribution function of a large random symmetric matrix tends to a distribution whose density function, when suitably normalized, has a semicircular graph. (This result is true for several models of random symmetric matrices: in particular, it is true if all entries on and below the diagonal are independent Gaussians with mean 0 and variance 1, with the entries above the diagonal determined by symmetry.) It also received a huge boost in 1973 as a result of an observation by Hugh Montgomery and Freeman Dyson that the correlation between pairs of nearby zeros of the Riemann zeta function appears to be the same, after normalization, as the correlation between pairs of eigenvalues of a random unitary or Hermitian matrices. (Roughly speaking, the pair correlation tells you what you expect the distribution of the remaining points to be given where one point is. In both cases there is a repulsive effect: zeros of the zeta function do not like to be too close to each other, and neither do eigenvalues of these random matrices.)

This paper concerns random unitary matrices. The eigenvalues of a unitary matrix lie on the unit circle in the complex plane, and the distribution is rotation invariant (since if one multiplies a random unitary matrix by a scalar of modulus 1, its distribution is unaffected). The main result of this paper is that it also satisfies a self-similarity property, which states, roughly, that the distribution of eigenvalues in an interval is similar to the distribution in the entire circle. To be more precise about it, suppose we take a random \(nm\times nm\) unitary matrix, take the eigenvalues of that matrix that have arguments in the interval \([-\pi/m,\pi/m]\), and raise each one to the power \(m\) (so that the interval \([-\pi/m,\pi/m]\) is stretched to the interval \([-\pi,\pi]\)). Then the resulting distribution of points will closely resemble the distribution of eigenvalues of a random \(n\times n\) unitary matrix.

Closeness is measured in the \(L_1\)-Wasserstein distance, which is defined as follows. If \(\mu\) and \(\nu\) are two probability distributions on a metric space \(M\), then their \(L_1\)-Wasserstein distance is the infimum of \(\mathbb E\, d(X,Y)\) over all pairs of random variables \(X\) and \(Y\) such that \(X\) has distribution \(\mu\) and \(Y\) has distribution \(\nu\). Intuitively, it is the total amount by which you have to “move” \(\mu\) to turn it into \(\nu\) if you do so as economically as possible.

This result proves a conjecture of Coram and Diaconis, in a slightly modified form. (In the original conjecture, one chooses not an interval of fixed length but an interval from one eigenvalue to the one \(n\) places along as you go round the circle, and that interval is stretched and randomly rotated.)

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