TY - JOUR

T1 - Universality for 1d Random Band Matrices

T2 - Sigma-Model Approximation

AU - Shcherbina, Mariya

AU - Shcherbina, Tatyana

N1 - Funding Information:
T. Shcherbina was supported in part by NSF Grant DMS-1700009.
Funding Information:
We are grateful to Yan Fyodorov for his suggestion of this particular model for the derivation of sigma-model approximation for RBM. TS would like to thank Tom Spencer for his explanation of the nature of sigma-model approximation and for many fruitful discussions without that this paper would never have been written. T. Shcherbina was supported in part by NSF Grant DMS-1700009.
Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2018/7/1

Y1 - 2018/7/1

N2 - 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.

AB - 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.

KW - Random band matrices

KW - Sigma-model approximation

KW - Transfer matrix approach

KW - Universality

UR - http://www.scopus.com/inward/record.url?scp=85049577689&partnerID=8YFLogxK

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U2 - 10.1007/s10955-018-1969-1

DO - 10.1007/s10955-018-1969-1

M3 - Article

AN - SCOPUS:85049577689

SN - 0022-4715

VL - 172

SP - 627

EP - 664

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 2

ER -