Universality for 1d Random Band Matrices: Sigma-Model Approximation

@article{Shcherbina2018UniversalityF1,
  title={Universality for 1d Random Band Matrices: Sigma-Model Approximation},
  author={Mariya Shcherbina and Tatyana Shcherbina},
  journal={Journal of Statistical Physics},
  year={2018},
  volume={172},
  pages={627-664}
}
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 $$W\times W$$W×W random Gaussian blocks (parametrized by $$j,k \in \Lambda =[1,n]^d\cap \mathbb {Z}^d$$j,k∈Λ=[1,n]d∩Zd) with a fixed entry’s variance $$J_{jk}=\delta _{j,k}W^{-1}+\beta \Delta _{j,k}W^{-2}$$Jjk=δj,kW-1+βΔj,kW-2… 

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