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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References

SHOWING 1-10 OF 28 REFERENCES

Characteristic Polynomials for 1D Random Band Matrices from the Localization Side

We study the special case of $${n\times n}$$n×n 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by $${J=(-W^2\triangle+1)^{-1}}$$J=(-W2▵+1)-1. Assuming

Universality of the Local Regime for the Block Band Matrices with a Finite Number of Blocks

TLDR
The Hermitian matrices are the special case of Wegner’s W-orbital models and it is proved universality of the local eigenvalue statistics of H_N$$HN for the energies|\lambda _0 |< \sqrt{2}$$|λ0|<2.

Bulk universality for generalized Wigner matrices

Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure νij with a subexponential decay. Let $${\sigma_{ij}^2}$$

Delocalization for a class of random block band matrices

We consider $$N\times N$$N×N Hermitian random matrices H consisting of blocks of size $$M\ge N^{6/7}$$M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment

Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

We consider Hermitian and symmetric random band matrices H in d ≥ 1 dimensions. The matrix elements Hxy, indexed by $${x,y \in \Lambda \subset \mathbb{Z}^d}$$, are independent, uniformly distributed

On the Second Mixed Moment of the Characteristic Polynomials of 1D Band Matrices

We consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of 1D Gaussian band matrices, i.e., of the Hermitian N × N matrices HN with independent Gaussian

Delocalization and Diffusion Profile for Random Band Matrices

We consider Hermitian and symmetric random band matrices H = (hxy) in $${d\,\geqslant\,1}$$d⩾1 dimensions. The matrix entries hxy, indexed by $${x,y \in (\mathbb{Z}/L\mathbb{Z})^d}$$x,y∈(Z/LZ)d, are

Transfer Matrix Approach to 1d Random Band Matrices: Density of States

We study the special case of $$n\times n$$n×n 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix $$J=(-W^2\triangle +1)^{-1}$$J=(-W2▵+1)-1.

Density of States for Random Band Matrices in Two Dimensions

We consider a two-dimensional random band matrix ensemble, in the limit of infinite volume and fixed but large band width W. For this model, we rigorously prove smoothness of the averaged density of

Scaling properties of localization in random band matrices: A sigma -model approach.

TLDR
It is proved that ${\mathit{b}}^{2}$/N, N being the matrix size, is the relevant scaling parameter when the mean value of diagonal elements increases linearly along the diagonal an extra scaling parameter arises.