Universality for 1d Random Band Matrices: SigmaModel Approximation
Abstract
The paper continues the development of the rigorous supersymmetric transfer matrix approach to the random band matrices started in (J Stat Phys 164:12331260, 2016; Commun Math Phys 351:10091044, 2017). We consider random Hermitian block band matrices consisting of W× W random Gaussian blocks (parametrized by j,k \in Λ =[1,n]^d\cap Z^d) with a fixed entry's variance J_{jk}=δ _{j,k}W^{1}+β Δ _{j,k}W^{2}, β >0 in each block. Taking the limit W→ ∞ with fixed n and β , we derive the sigmamodel approximation of the second correlation function similar to Efetov's one. Then, considering the limit β , n→ ∞, we prove that in the dimension d=1 the behaviour of the sigmamodel approximation in the bulk of the spectrum, as β ≫ n, is determined by the classical WignerDyson statistics.
 Publication:

Journal of Statistical Physics
 Pub Date:
 July 2018
 DOI:
 10.1007/s1095501819691
 arXiv:
 arXiv:1802.03813
 Bibcode:
 2018JSP...172..627S
 Keywords:

 Random band matrices;
 Sigmamodel approximation;
 Universality;
 Transfer matrix approach;
 Mathematical Physics
 EPrint:
 38 pp