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

Mariya Shcherbina, Tatyana Shcherbina

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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× W random Gaussian blocks (parametrized by j, k∈ Λ = [1 , n] d∩ Zd) with a fixed entry’s variance Jjk= δj,kW- 1+ βΔ j,kW- 2, β> 0 in each block. Taking the limit W→ ∞ with fixed n and β, we derive the sigma-model 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 sigma-model approximation in the bulk of the spectrum, as β≫ n, is determined by the classical Wigner–Dyson statistics.

Original languageEnglish (US)
Pages (from-to)627-664
Number of pages38
JournalJournal of Statistical Physics
Issue number2
StatePublished - Jul 1 2018

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Random band matrices
  • Sigma-model approximation
  • Transfer matrix approach
  • Universality


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