Averaging Fluctuations in Resolvents of Random Band Matrices
By L. Erdős, Antti Knowles, Horng-Tzer Yau. 2012
146 citations recorded by the source corpus.
We consider a general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint. We establish precise high-probability bounds on the averages of arbitrary monomials in the resolvent matrix entries. Our results generalize the previous results of Erdős et al. (Ann Probab, arXiv:1103.1919, 2013; Commun Math Phys, arXiv:1103.3869, 2013; J Combin 1(2):15–85, 2011) which constituted a key step in the proof of the local semicircle law with optimal error bound in mean-field random matrix models. Our bounds apply to random band matrices and improve previous estimates from order 2 to order 4 in the cases relevant to applications. In particular, they lead to a proof of the diffusion approximation for the magnitude of the resolvent of random band matrices. This, in turn, implies new delocalization bounds on the eigenvectors. The applications are presented in a separate paper (Erdős et al., arXiv:1205.5669, 2013).
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/c2c118302274f64bb2a72b7e6d19d8637e129d58
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Citations in the corpus
- Singular vector and singular subspace distribution for the matrix denoising model (2018, methodology): “We also adopt the notion of stochastic domination introduced in [29].”
- Edge universality for deformed Wigner matrices (2014, methodology): “A slightly different form was first used in [16].”
- Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphs (2017, methodology): “Higher order corrections to the self-consistent equations or the “fluctuation averaging lemma” were also constructed in [4, 15, 17, 46].”
- A Dynamical Approach to Random Matrix Theory (2017, methodology): “Quadratic functionals were considered in [62,69]; the current formulation is taken from [52].”
- Statistical inference for principal components of spiked covariance matrices (2020, background): “The notion of stochastic domination was introduced in [31], which provides a precise statement of the form “XN is bounded by YN up to a small power of N with high probability”.”
- On the principal components of sample covariance matrices (2014, methodology): “A systematic representation of a large family of random variables in terms of polynomials was first given in [18], and was combined with a parity argument in [10].”
- Spiked separable covariance matrices and principal components (2019, methodology): “We will use the following notion of stochastic domination, which was first introduced in [19] and subsequently used in many works on random matrix theory, such as [7, 8, 9, 20, 21, 30].”