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