CLT for non-Hermitian random band matrices with variance profiles
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We show that the fluctuations of the linear eigenvalue statistics of a non-Hermitian random band matrix of bandwidth b_n with a continuous variance profile w_ν(x) converges to a N(0,σ_f^2(ν)), where ν=_n→∞(2b_n/n). We obtain an explicit formula for σ_f^2(ν), which depends on the test function, and w_ν. When ν=1, the formula is consistent with Rider, and Silverstein (2006). We also compute an explicit formula for σ_f^2(0). We show that σ_f^2(ν)→σ_f^2(0) as ν↓ 0.
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