Spectrum of SYK model II: Central limit theorem

In our previous paper [R. Feng, G. Tian and D. Wei, Spectrum of SYK model, Peking Math. J. 2 (2019) 41–70], we derived the almost sure convergence of the global density of eigenvalues of random matrices of the SYK model. In this paper, we will prove the central limit theorem for the linear statistics of eigenvalues of the SYK model and compute its variance.

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CENTRAL LIMIT THEOREM FOR LINEAR STATISTICS OF EIGENVALUES OF BAND RANDOM MATRICES

Random Matrices Theory and Application ◽

10.1142/s2010326313500093 ◽

2013 ◽

Vol 02 (04) ◽

pp. 1350009 ◽

Cited By ~ 9

Author(s):

LINGYUN LI ◽

ALEXANDER SOSHNIKOV

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Random Matrices ◽

Central Limit ◽

Test Functions ◽

Linear Statistics ◽

The Central Limit Theorem ◽

Linear Statistics Of Eigenvalues

We prove the Central Limit Theorem for linear statistics of the eigenvalues of band random matrices provided [Formula: see text] and test functions are sufficiently smooth.

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On the law of addition of random matrices: Covariance and the central limit theorem for traces of resolvent

Probability and Mathematical Physics - CRM Proceedings and Lecture Notes ◽

10.1090/crmp/042/21 ◽

2007 ◽

pp. 399-416 ◽

Cited By ~ 3

Author(s):

Leonid Pastur ◽

V. Vasilchuk

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Random Matrices ◽

Central Limit ◽

The Central Limit Theorem ◽

The Law

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On the fluctuations of eigenvalues of multiplicative deformed unitary invariant ensembles

Random Matrices Theory and Application ◽

10.1142/s2010326316500076 ◽

2016 ◽

Vol 05 (02) ◽

pp. 1650007 ◽

Cited By ~ 1

Author(s):

Vladimir Vasilchuk

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Random Matrices ◽

Central Limit ◽

Counting Measure ◽

Linear Eigenvalue Statistics ◽

The Central Limit Theorem ◽

Eigenvalue Statistics ◽

Leading Term

We consider the ensemble of [Formula: see text] random matrices [Formula: see text], where [Formula: see text] and [Formula: see text] are non-random, unitary, having the limiting Normalized Counting Measure (NCM) of eigenvalues, and [Formula: see text] is unitary, uniformly distributed over [Formula: see text]. We find the leading term of the covariance of traces of resolvent of [Formula: see text] and establish the Central Limit Theorem for sufficiently smooth linear eigenvalue statistics of [Formula: see text] as [Formula: see text].

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Free probability

10.1093/oso/9780198797319.003.0003 ◽

2018 ◽

Author(s):

Alice Guionnet

Keyword(s):

Brownian Motion ◽

Central Limit Theorem ◽

Limit Theorem ◽

Random Matrices ◽

Central Limit ◽

Differential Calculus ◽

Free Probability ◽

Integration Theory ◽

The Central Limit Theorem ◽

Planar Maps

Free probability was introduced by D. Voiculescu as a theory of noncommutative random variables (similar to integration theory) equipped with a notion of freeness very similar to independence. In fact, it is possible in this framework to define the natural ‘free’ counterpart of the central limit theorem, Gaussian distribution, Brownian motion, stochastic differential calculus, entropy, etc. It also appears as the natural setup for studying large random matrices as their size goes to infinity and hence is central in the study of random matrices as their size go to infinity. In this chapter the free probability framework is introduced, and it is shown how it naturally shows up in the random matrices asymptotics via the so-called ‘asymptotic freeness’. The connection with combinatorics and the enumeration of planar maps, including loop models, are discussed.

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