[Paper Review] Characteristic polynomials of modified permutation matrices at microscopic scale

Valentin Bahier|arXiv (Cornell University)|Jan 31, 2018

Analytic Number Theory Research22 references4 citations

TL;DR

This paper studies the microscopic scaling limit of characteristic polynomials of random permutation matrices and their modifications, where 1-entries are replaced by i.i.d. uniform variables on the unit circle. Under Ewens measures and appropriate normalization, the rescaled characteristic polynomial converges almost surely to an explicit entire function, extending results from the Circular Unitary Ensemble to permutation-based random matrices with a coupling method inspired by virtual isometries.

ABSTRACT

We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation group. We also look at some modifications of permutation matrices where the entries equal to one are replaced by i.i.d uniform variables on the unit circle. Once appropriately normalized and scaled, we show that the characteristic polynomial converges in distribution on every compact subset of $\\mathbb{C}$ to an explicit limiting entire function, when the size of the matrices goes to infinity. Our findings can be related to results by Chhaibi, Najnudel and Nikeghbali on the limiting characteristic polynomial of the Circular Unitary Ensemble.

Motivation & Objective

• To analyze the asymptotic behavior of the characteristic polynomial of random permutation matrices under conjugation-invariant measures, including Ewens measures.
• To extend results from the Circular Unitary Ensemble to permutation matrices by studying the microscopic scaling of their characteristic polynomials.
• To establish almost sure convergence of the rescaled characteristic polynomial to a limiting entire function using a novel coupling method.
• To generalize the limiting behavior to modified permutation matrices with i.i.d. unit-circular entries instead of 1s.
• To characterize the growth and analytic properties of the limiting entire function in terms of cycle lengths and uniform random variables on the unit circle.

Proposed method

• Introduces a coupling method for generating sequences of virtual permutations and modified permutations using i.i.d. uniform variables on the unit circle.
• Defines the rescaled characteristic polynomials $ ilde{ heta}_n(z) = rac{ ilde{Z}_n(e^{2i au z/n})}{ ilde{Z}_n(1)} $ and $ heta_{n,eta}(z) = rac{Z_n(e^{2i au(z/n + eta)})}{Z_n(e^{2i aueta})} $ for $ z o ext{compact set in } bC $, with $ eta $ irrational.
• Expresses the rescaled polynomials in terms of normalized cycle lengths $ y_j^{(n)} $ and i.i.d. unit-circular variables $ u_j $, yielding $ ilde{ heta}_n(z) = extstyleigprod_{j: ext{cycle length } e 0} rac{e^{2i au z y_j^{(n)}} - u_j}{1 - u_j} $.
• Uses the convergence of normalized cycle lengths to a Poisson-Dirichlet process and the coupling of virtual isometries to achieve almost sure convergence.
• Applies moderate deviation estimates and product estimates over zeros $ w_k $ of the limiting function to control growth and analyticity.
• Establishes bounds on the growth of the limiting function using $ | ilde{ heta}_ y{infty}(z)| le ext{e}^{C|z| ext{log}(2+|z|)} $ and similar for $ heta_{ y{infty},eta}(z) $.

Experimental results

Research questions

• RQ1 Does the characteristic polynomial of a random permutation matrix under Ewens measure converge to a limiting entire function at the microscopic scale?
• RQ2 How does replacing 1-entries in permutation matrices with i.i.d. uniform variables on the unit circle affect the scaling limit of the characteristic polynomial?
• RQ3 Can a coupling method based on virtual isometries ensure almost sure convergence of the rescaled characteristic polynomial?
• RQ4 What are the growth and analytic properties of the limiting entire function arising from the rescaled characteristic polynomial?
• RQ5 How does the limiting behavior depend on the irrational parameter $ eta $ in the scaling $ z/n + eta $?

Key findings

• The rescaled characteristic polynomial $ ilde{ heta}_n(z) $ converges almost surely to an entire function $ ilde{ heta}_ y{infty}(z) = extstyleigprod_{j=1}^ y{infty} rac{e^{2i au z y_j} - u_j}{1 - u_j} $ as $ n o y{infty} $, where $ (y_j) $ are normalized cycle lengths and $ u_j $ are i.i.d. uniform on the unit circle.
• The limiting function $ ilde{ heta}_ y{infty}(z) $ satisfies the growth bound $ | ilde{ heta}_ y{infty}(z)| le ext{e}^{C|z| ext{log}(2+|z|)} $ almost surely for some random $ C > 0 $.
• For the standard permutation matrix case, the rescaled polynomial $ heta_{n,eta}(z) $ converges almost surely to $ heta_{ y{infty},eta}(z) = extstyleigprod_{j: ext{cycle length } e 0} rac{e^{2i au(z/n + eta) ilde{ au}_j} - 1}{e^{2i aueta ilde{ au}_j} - 1} $, with $ eta $ irrational.
• The limiting function $ heta_{ y{infty},eta}(z) $ satisfies $ | heta_{ y{infty},eta}(z)| le C_ y{ y{eps}} ext{e}^{( y{eps} + 2 au(y_0 + (1-y_0)t_eta))|z|} $ for all $ y{eps} > 0 $, where $ t_eta = rac{1}{2 ext{sin}( aueta)} $.
• The zeros $ w_k $ of the limiting function satisfy $ w_k = rac{k}{1 - y_0} + y{O}(k^{1/2 + y{eps}}) $ almost surely for any $ y{eps} > 0 $.
• The convergence of $ ilde{ heta}_n(z) $ to $ ilde{ heta}_ y{infty}(z) $ holds in law on compact subsets of $ bC $, and the coupling ensures almost sure convergence, extending results from the CUE ensemble to permutation-based models.