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  1. (, Journal of statistical physics)
    The paper continues (Shcherbina and Shcherbina in Commun Math Phys 351:1009–1044, 2017); Shcherbina in Commun Math Phys 328:45–82, 2014) which study the behaviour of second correlation function of characteristic polynomials of the special case of $$n \times n$$ one-dimensional Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix $$𝐽=(-W^2\triangle+1)^{-1}$$. Applying the transfer matrix approach, we study the case when the bandwidth W is proportional to the threshold $$\sqrt{n}$$ 
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  2. ; (, Proceedings of the International Congress of Mathematicians 2018)
    We discuss an application of the transfer operator approach to the analysis of the different spectral characteristics of 1d random band matrices (correlation functions of characteristic polynomials, density of states, spectral correlation functions). We show that when the bandwidth $$W$$ crosses the threshold $$W=N^{1/2}$$, the model has a kind of phase transition (crossover), whose nature can be explained by the spectral properties of the transfer operator. 
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  3. ; (, Journal of Statistical Physics)
    The paper continues the development of the rigorous supersymmetric transfer matrix approach to the random band matrices started in (J Stat Phys 164:1233–1260, 2016; Commun Math Phys 351:1009–1044, 2017). We consider random Hermitian block band matrices consisting of $$W\times W$$ random Gaussian blocks (parametrized by $$j,k \in\Lambda=[1,n]^d\cap \mathbb{Z}^d$$) with a fixed entry's variance $$J_{jk}=\delta_{j,k}W^{-1}+\beta\Delta_{j,k}W^{-2}$$, $$\beta>0$$ in each block. Taking the limit $$W\to\infty$$ with fixed $$n$$ and $$\beta$$, we derive the sigma-model approximation of the second correlation function similar to Efetov's one. Then, considering the limit $$\beta, n\to\infty$$, we prove that in the dimension $d=1$ the behaviour of the sigma-model approximation in the bulk of the spectrum, as $$\beta\gg n$$, is determined by the classical Wigner -- Dyson statistics. 
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