Almost-optimal bulk regularity conditions in the CLT for Wigner matrices

Landon, Benjamin; Sosoe, Philippe

doi:10.48550/arxiv.2204.03419

Cited by 3 publications

(5 citation statements)

References 41 publications

(91 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note too the work of Shcherbina [144] for an independent evaluation of the limit in (3.54) in the real case for f = g, which gives a functional form generalising the final equality in (3.4). A comprehensive study of conditions on f for which this formula is valid has recently been given by Landon and Sosoe [115]; this work also reviews earlier work along these lines as part of the Introduction section. In the complex case Bao and Xie [16] remove the requirement of [12] that |x r ij | 2 = |x i ij | 2 ; the covariance formula now depends on the parameter Φ…”

Section: Wigner Matricesmentioning

confidence: 91%

A review of exact results for fluctuation formulas in random matrix theory

Forrester¹

2022

Preprint

View full text Add to dashboard Cite

Covariances and variances of linear statistics of a point process can be written as integrals over the truncated two-point correlation function. When the point process consists of the eigenvalues of a random matrix ensemble, there are often large N universal forms for this correlation after smoothing, which results in particularly simple limiting formulas for the fluctuation of the linear statistics. We review these limiting formulas, derived in the simplest cases as corollaries of explicit knowledge of the truncated two-point correlation. One of the large N limits is to scale the eigenvalues so that limiting support is compact, and the linear statistics vary on the scale of the support. This is a global scaling. The other, where a thermodynamic limit is first taken so that the spacing between eigenvalues is of order unity, and then a scale imposed on the test functions so they are slowly varying, is the bulk scaling. The latter was already identified as a probe of random matrix characteristics for quantum spectra in the pioneering work of Dyson and Mehta.

show abstract

Section: Wigner Matricesmentioning

confidence: 91%

A review of exact results for fluctuation formulas in random matrix theory

Forrester¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Example 1. 16 Whenever κ 4 = σ = ω 2 = 0 and one argument of m GU E [•|•] is a singleton set, only the fourth line of (1.28) gives a non-zero contribution. Rewriting this term using (1.16) (cf.…”

Section: Note That the Recursion For M[•|•]mentioning

confidence: 99%

“…, f k+ ∈ H +1 (R). The functions 16) can be computed recursively and only depend on the underlying permutation resp. marked partition, the functions f 1 , .…”

Section: General Test Functions and Applications To Free Probabilitymentioning

confidence: 99%

Fluctuation Moments for Regular Functions of Wigner Matrices

Reker

2024

Math Phys Anal Geom

View full text Add to dashboard Cite

We compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices. Restricting to polynomials, our formulas reproduce recent results of Male et al. (Random Matrices Theory Appl. 11(2):2250015, 2022), showing that the underlying combinatorics of non-crossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. The formulas obtained further characterize the variance in the functional central limit theorem given in the recent companion paper (Reker in Preprint, arXiv:2204.03419, 2023). and thus allow identifying the fluctuation around the thermal value in certain thermalization problems.

show abstract

“…Dynamical approach to multi-resolvent local laws. Multi-resolvent local laws have recently been derived in various situations; they are necessary to prove the central limit theorem for linear eigenvalue statistics [86,95,112,79,90,89,52,55,58,91,93,117,106,107] and the Eigenstate Thermalisation Hypothesis [54,47,1]. Their canonical proof is to derive an approximate self-consistent equation, e.g.…”

mentioning

confidence: 99%

“…The method of characteristic flow has been effectively used for local laws and closely related quantities in various models in [81,2,89,3,91,35,34,92] but only for a single resolvent situation where it eventually leads to a linear equation thanks to the Ward identity. Two or more resolvents with dynamical method were considered only in [58,Theorem 3.3], [35,Proposition 4.5], [47,Proposition 3.3] and very recently in [118,Lemma 4.2].…”

mentioning

confidence: 99%

Directional extremal statistics for Ginibre eigenvalues

Cipolloni

Erdős

Schröder

et al. 2022

Journal of Mathematical Physics

View full text Add to dashboard Cite

We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme eigenvalues form a Poisson point process as the dimension asymptotically tends to infinity. In the complex case, these facts have already been established by Bender [Probab. Theory Relat. Fields 147, 241 (2010)] and in the real case by Akemann and Phillips [J. Stat. Phys. 155, 421 (2014)] even for the more general elliptic ensemble with a sophisticated saddle point analysis. The purpose of this article is to give a very short direct proof in the Ginibre case with an effective error term. Moreover, our estimates on the correlation kernel in this regime serve as a key input for accurately locating [Formula: see text] for any large matrix X with i.i.d. entries in the companion paper [G. Cipolloni et al., arXiv:2206.04448 (2022)].

show abstract

Almost-optimal bulk regularity conditions in the CLT for Wigner matrices

Cited by 3 publications

References 41 publications

A review of exact results for fluctuation formulas in random matrix theory

A review of exact results for fluctuation formulas in random matrix theory

Fluctuation Moments for Regular Functions of Wigner Matrices

Directional extremal statistics for Ginibre eigenvalues

Contact Info

Product

Resources

About