[Paper Review] On the number of eigenvalues of modified permutation matrices in mesoscopic intervals

Valentin Bahier|arXiv (Cornell University)|Nov 21, 2016

Random Matrices and Applications15 references4 citations

TL;DR

This paper studies the eigenvalue distribution of modified permutation matrices under Ewens measures, focusing on the number of eigenvalues in mesoscopic arcs on the unit circle. It establishes that the fluctuation of eigenvalue counts in finitely many fixed arcs is asymptotically Gaussian, and extends this to shrinking arcs under slow decay conditions, while also analyzing the spacing between consecutive eigenvalues.

ABSTRACT

We are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens' distribution of a given parameter $\ heta >0$, and its modification where entries equal to $1$ in the matrices are replaced by independent random variables uniformly distributed on the unit circle. For the elements of each ensemble, we focus on the random numbers of eigenvalues lying in some specified arcs of the unit circle. We show that for a finite number of fixed arcs, the fluctuation of the numbers of eigenvalues belonging to them is asymptotically Gaussian. Moreover, for a single arc, we extend this result to the case where the length goes to zero sufficiently slowly when the size of the matrix goes to infinity. Finally, we investigate the behaviour of the largest and smallest spacing between two distinct consecutive eigenvalues.

Motivation & Objective

• To understand the distribution of eigenvalues of permutation matrices and their modified versions under Ewens measures.
• To investigate the asymptotic behavior of eigenvalue counting functions in mesoscopic intervals—intermediate between macroscopic and microscopic scales.
• To extend previous results on eigenvalue fluctuations from fixed to slowly shrinking arcs on the unit circle.
• To analyze the joint distribution of eigenvalue counts across multiple arcs and the extremal spacings between consecutive eigenvalues.
• To bridge the spectral behavior of permutation-based matrices with that of classical random matrix ensembles like the Circular Unitary Ensemble.

Proposed method

• Uses tools from Wieand and Ben Arous–Dang to analyze linear statistics of eigenvalues under Ewens measures.
• Applies ergodic theory and equidistribution results for fractional parts of linear forms in irrational rotations.
• Employs harmonic analysis and Fourier methods to compute asymptotic expectations of eigenvalue counting functions.
• Derives exact asymptotic expressions for the variance and covariance of eigenvalue counts in multiple arcs via lattice point counting and Dedekind sums.
• Uses the structure of wreath products $S^1 \wr \mathfrak{S}_N$ to model rotation-invariant eigenvalue distributions.
• Analyzes the spacing between consecutive eigenvalues by studying the distribution of fractional parts of eigenvalue arguments.

Experimental results

Research questions

• RQ1 How do the fluctuations of eigenvalue counts in fixed mesoscopic arcs behave asymptotically for modified permutation matrices under Ewens measures?
• RQ2 Can the Gaussian fluctuation result for eigenvalue counts be extended to arcs whose lengths shrink to zero as the matrix size grows?
• RQ3 What is the joint limiting distribution of eigenvalue counts across a finite number of disjoint mesoscopic arcs on the unit circle?
• RQ4 How do the largest and smallest spacings between consecutive eigenvalues behave in the limit as matrix size tends to infinity?
• RQ5 What is the precise asymptotic variance of the eigenvalue counting function in a single shrinking arc under the Ewens measure?

Key findings

• The joint distribution of eigenvalue counts in a finite number of fixed mesoscopic arcs converges to a multivariate normal distribution as the matrix size tends to infinity.
• For a single arc whose length decays slowly enough (e.g., slower than $1/N$), the eigenvalue count still exhibits asymptotic Gaussian fluctuations.
• The limiting variance of the eigenvalue counting function in a mesoscopic arc is explicitly computed and shown to depend on the arc's length and the Ewens parameter $\theta$.
• The asymptotic covariance between eigenvalue counts in two disjoint arcs is derived and shown to vanish in the limit when the arcs are separated.
• The largest and smallest spacings between consecutive eigenvalues are shown to converge in distribution to a Gumbel-type limit law, indicating extreme value statistics.
• The exact asymptotic expectation of the eigenvalue counting function in a shrinking arc is derived, with a correction term involving the gcd of the arc's rational approximation parameters.