On the spectral distribution of large weighted random regular graphs

Goldmakher, Leo; Khoury, Cap; Miller, Steven J.; Ninsuwan, Kesinee

doi:10.1142/s2010326314500154

Cited by 8 publications

(8 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

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

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

“…In this paper we concentrate on the density of states, though there has been significant progress in recent years on the spacings between normalized eigenvalues (see for example [ERSY,ESY,TV1,TV2]). The limiting spectral measure has been computed for many ensembles of patterned or structured matrices, including band, circulant, Hankel and Toeplitz matrices, as well as adjacency matrices of d-regular graphs [BBDS,BasBo1,BasBo2,BanBo,BCG,BHS1,BHS2,BM,BDJ,GKMN,HM,JMP,Kar,KKMSX,LW,McK,Me,Sch]. Many of these papers show that when a classical ensemble is modified by enforcing particular, usually linear, relations among the entries, new limiting behavior emerges.…”

Section: Introductionmentioning

confidence: 99%

Spherical Matrix Ensembles

Kopp,

Miller

2015

Preprint

Self Cite

View full text Add to dashboard Cite

The spherical orthogonal, unitary, and symplectic ensembles (SOE/SUE/SSE) S β (N, r) consist of N × N real symmetric, complex hermitian, and quaternionic self-adjoint matrices of Frobenius norm r, made into a probability space with the uniform measure on the sphere. For each of these ensembles, we determine the joint eigenvalue distribution for each N , and we prove the empirical spectral measures rapidly converge to the semicircular distribution as N → ∞. In the unitary case (β = 2), we also find an explicit formula for the empirical spectral density for each N .

show abstract

On the spectral distribution of large weighted random regular graphs

Cited by 8 publications

References 37 publications

The limiting spectral measure for an ensemble of generalized checkerboard matrices

The limiting spectral measure for an ensemble of generalized checkerboard matrices

Spectral distributions of periodic random matrix ensembles

Spherical Matrix Ensembles

Contact Info

Product

Resources

About