Characteristic polynomials for 1D random band matrices from the localization side

Abstract

We study the special case of $n\times n$ 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by $J=(-W^2△+1{)}^{-1}$. Assuming that the band width $W\ll \sqrt{n}$, we prove that the limit of the normalized second mixed moment of characteristic polynomials (as $W,n\to \infty$) is equal to one, and so it does not coincides with those for GUE. This complements the previous result of T. Shcherbina and proves the expected crossover for 1D Hermitian random band matrices at $W\sim \sqrt{n}$ on the level of characteristic polynomials.

30 p

References in corpus (4)

• Universality for a class of random band matrices
• On the correlation functions of the characteristic polynomials of the sparse hermitian random matrices
• Delocalization for a class of random block band matrices
• On the correlation functions of the characteristic polynomials of the hermitian sample covariance ensemble

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• Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble
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• On characteristic polynomials for a generalized chiral random matrix ensemble with a source
• Random band matrices in the delocalized phase, III: Averaging fluctuations
• On the Correlation Functions of the Characteristic Polynomials of Non-Hermitian Random Matrices with Independent Entries
• Anderson-Bernoulli Localization on the 3D lattice and discrete unique continuation principle
• Delocalization and quantum diffusion of random band matrices in high dimensions II: $T$-expansion
• On the local eigenvalue statistics for random band matrices in the localization regime
• The least singular value of the general deformed Ginibre ensemble
• Characteristic polynomials for random band matrices near the threshold
• Universal characteristics of deep neural network loss surfaces from random matrix theory
• A supersymmetric hierarchical model for weakly disordered $3d$ semimetals
• Statistical field theory of random graphs with prescribed degrees
• Transfer matrix approach for the real symmetric 1D random band matrices
• Limiting eigenvalue distribution of the general deformed Ginibre ensemble
• Evidence of the Poisson/Gaudin-Mehta phase transition for banded matrices on global scales