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Universality for 1d random band matrices: sigma-model approximation
- Publication Year :
- 2018
-
Abstract
- 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.<br />Comment: 38 pp
- Subjects :
- Mathematical Physics
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1802.03813
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s10955-018-1969-1