Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices

Jackson, Steven Glenn; Miller, Steven J.; Pham, Thuy Linh

doi:10.1007/s10959-010-0311-x

Cited by 12 publications

(8 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently there has been much interest in studying highly structured sub-ensembles of the family of real symmetric matrices, where new limiting behavior emerges. Examples include band matrices, circulant matrices, random abelian G-circulant matrices, adjacency matrices associated to d-regular graphs, and Hankel and Toeplitz matrices, among others [BasBo2,BasBo1,BanBo,BCG,BHS1,BHS2,BM,BDJ,GKMN,HM,JMP,Kar,KKMSX,LW,MMS,McK,Me,Sch]. Two particularly interesting cases are the Toeplitz [BDJ, HM] and singly palindromic Toeplitz ensemble [MMS], which we now generalize (though our arguments would follow through with only minor changes for other structured ensembles).…”

mentioning

confidence: 77%

Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles

Beckwith¹,

Luo²,

Miller³

et al. 2011

Preprint

Self Cite

View full text Add to dashboard Cite

The study of the limiting distribution of eigenvalues of N × N random matrices as N → ∞ has many applications, including nuclear physics, number theory and network theory. One of the most studied ensembles is that of real symmetric matrices with independent entries drawn from identically distributed nice random variables, where the limiting rescaled spectral measure is the semi-circle. Studies have also determined the limiting rescaled spectral measures for many structured ensembles, such as Toeplitz and circulant matrices. These systems have very different behavior; the limiting rescaled spectral measures for both have unbounded support. Given a structured ensemble such that (i) each random variable occurs o(N ) times in each row of matrices in the ensemble and (ii) the limiting rescaled spectral measure µ exists, we introduce a parameter to continuously interpolate between these two behaviors. We fix a p ∈ [1/2, 1] and study the ensemble of signed structured matrices by multiplying the (i, j) th and (j, i) th entries of a matrix by a randomly chosen ǫ ij ∈ {1, −1}, with Prob(ǫ ij = 1) = p (i.e., the Hadamard product). For p = 1/2 we prove that the limiting signed rescaled spectral measure is the semi-circle. For all other p, we prove the limiting measure has bounded (resp., unbounded) support if µ has bounded (resp., unbounded) support, and converges to µ as p → 1. Notably, these results hold for Toeplitz and circulant matrix ensembles.The proofs are by Markov's Method of Moments. The analysis of the 2k th moment for such distributions involves the pairings of 2k vertices on a circle. The contribution of each pairing in the signed case is weighted by a factor depending on p and the number of vertices involved in at least one crossing. These numbers are of interest in their own right, appearing in problems in combinatorics and knot theory. The number of configurations with no vertices involved in a crossing is well-studied, and are the Catalan numbers. We discover and prove similar formulas for configurations with 4, 6, 8 and 10 vertices in at least one crossing. We derive a closed-form expression for the expected value and determine the asymptotics for the variance for the number of vertices in at least one crossing. As the variance converges to 4, these results allow us to deduce properties of the limiting measure.

show abstract

mentioning

confidence: 77%

Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles

Beckwith¹,

Luo²,

Miller³

et al. 2011

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Besides the more well-known families such as the Gaussian Orthogonal, Unitary and Symplectic Ensembles, many other special ensembles have been studied; see for example [Bai,BasBo1,BasBo2,BanBo,BLMST,BCG,BHS1,BHS2,BM,BDJ,GKMN,HM,JMRR,JMP,Kar,KKMSX,LW,MMS,MNS,MSTW,McK,Me,Sch], where the additional structures on the entries of the matrices lead to different behaviors of the eigenvalues in the limit.…”

Section: Introductionmentioning

confidence: 99%

The limiting spectral measure for an ensemble of generalized checkerboard matrices

Chen

Lin

Miller

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

Random matrix theory successfully models many systems, from the energy levels of heavy nuclei to zeros of L-functions. While most ensembles studied have continuous spectral distribution, Burkhardt et al introduced the ensemble of k-checkerboard matrices, a variation of Wigner matrices with entries in generalized checkerboard patterns fixed and constant. In this family, N − k of the eigenvalues are of size O( √ N ) and were called bulk while the rest are tightly contrained around a multiple of N and were called blip.We extend their work by allowing the fixed entries to take different constant values. We can construct ensembles with blip eigenvalues at any multiples of N we want with any multiplicity (thus we can have the blips occur at sequences such as the primes or the Fibonaccis). The presence of multiple blips creates technical challenges to separate them and to look at only one blip at a time. We overcome this by choosing a suitable weight function which allows us to localize at each blip, and then exploiting cancellation to deal with the resulting combinatorics to determine the average moments of the ensemble; we then apply standard methods from probability to prove that almost surely the limiting distributions of the matrices converge to the average behavior as the matrix size tends to infinity. For blips with just one eigenvalue in the limit we have convergence to a Dirac delta spike, while if there are k eigenvalues in a blip we again obtain hollow k × k GOE behavior.

show abstract

“…Besides studying these classical ensembles, a substantial bulk of the random matrix theory literature is devoted to the eigenvalue distributions of various special "patterned" ensembles with often non-semicircular limiting spectral distribution. These may consist of Toeplitz, Hankel or various other types of matrices, for which there are additional restrictions on the entries [Bai,BasBo1,BasBo2,BanBo,BLMST,BCG,BHS1,BHS2,BM,BDJ,GKMN,HM,JMRR,JMP,Kar,KKMSX,LW,MMS,MNS,MSTW,McK,Me,Sch].…”

Section: Introductionmentioning

confidence: 99%

Spectral distributions of periodic random matrix ensembles

Peski

2019

Random Matrices: Theory Appl.

View full text Add to dashboard Cite

Kologlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as k-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an k × k Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Kologlu-Kopp-Miller, real-symmetric ensembles with periodic diagonals always have limiting spectral distribution equal to the eigenvalue distribution of a finite Hermitian ensemble with Gaussian entries which is a 'complex version' of a k × k submatrix of the ensemble. We also prove an essentially algebraic relation between certain periodic finite Hermitian ensembles with Gaussian entries, and the previous result may be seen as an asymptotic version of this for real-symmetric ensembles. The proofs show that this general correspondence between periodic random matrix ensembles and finite complex Hermitian ensembles is elementary and combinatorial in nature.

show abstract

Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices

Cited by 12 publications

References 23 publications

Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles

Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles

The limiting spectral measure for an ensemble of generalized checkerboard matrices

Spectral distributions of periodic random matrix ensembles

Contact Info

Product

Resources

About