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Localization of One-Dimensional Random Band Matrices
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Localization of One-Dimensional Random Band Matrices
abstract
We consider a general class of $n\times n$ random band matrices with bandwidth $W$. When $W^2\ll n$, we prove that with high probability the eigenvectors of such matrices are localized and decay exponentially at the sharp scale $W^2$. Combined with the delocalization results of Yau and Yin [arXiv:2501.01718] and of Erd\H{o}s and Riabov [arXiv:2506.06441], this establishes the conjectured localization-delocalization transition for a large class of random band matrices.
Forward citations
Cited by 5 Pith papers
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Localization Lengths of Power-Law Random Band Matrices
The paper establishes rigorous lower bounds on eigenvector localization lengths for power-law random band matrices in four regimes of the decay exponent α, verifying a physical conjecture via new resolvent techniques.
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On a Rosenzweig-Porter-type model
Uniform single- and two-resolvent local laws for arbitrary deformations of Wigner matrices yield ETH and localization profiles for all λ and H0.
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Gradual eigenvector ergodization in coupled Ginibre matrices
Explicit large-N asymptotic formula for gradual eigenvector ergodization in two coupled Ginibre matrices, plus vanishing of eigenvalue density at the origin beyond critical scaled coupling |tilde c|=1.
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Characteristic polynomials of non-Hermitian random band matrices near the threshold
The second correlation function of characteristic polynomials for non-Hermitian random band matrices is studied asymptotically in the critical regime W proportional to sqrt(N) as N and W tend to infinity.
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