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

  title={Universality for 1d Random Band Matrices: Sigma-Model Approximation},
  author={Mariya Shcherbina and Tatyana Shcherbina},
  journal={Journal of Statistical Physics},
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… 

Universality for 1d Random Band Matrices

It is proved that the behaviour of the second correlation function of such matrices in the bulk of the spectrum, as $W\gg \sqrt{N}$, is determined by the Wigner -- Dyson statistics.

Delocalization and Quantum Diffusion of Random Band Matrices in High Dimensions II: T-expansion

We consider Green's functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(\mathbb Z/L\mathbb Z)^d$. The entries $h_{xy}=\bar h_{yx}$ of $H$ are

Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

We consider an elastic manifold of internal dimension d and length L pinned in a N dimensional random potential and confined by an additional parabolic potential of curvature $$\mu $$ μ . We are

Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

We consider Hermitian random band matrices H = (hxy) on the d-dimensional lattice (Z/LZ)d, where the entries hxy = hyx are independent centered complex Gaussian random variables with variances sxy =

Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite range initial rates. Our proof has two main ingredients. The first is a direct

Delocalization and Continuous Spectrum for Ultrametric Random Operators

This paper studies the delocalized regime of an ultrametric random operator whose independent entries have variances decaying in a suitable hierarchical metric on $${\mathbb {N}}$$N. When the decay

Finite-rank complex deformations of random band matrices: sigma-model approximation

We study the distribution of complex eigenvalues z1, . . . , zN of random Hermitian N ×N block band matrices with a complex deformation of a finite rank. Assuming that the width of the bandW grows

Dynamical Localization for Random Band Matrices up to $W\ll N^{1/4}$

We consider a large class of N ×N Gaussian random band matrices with band-width W , and prove that for W ≪ N they exhibit Anderson localization at all energies. To prove this result, we rely on the

PR ] 4 F eb 2 01 9 The density of states of 1 D random band matrices via a supersymmetric transfer operator

Recently, T. and M. Shcherbina proved a pointwise semicircle law for the density of states of one-dimensional Gaussian band matrices of large bandwidth. The main step of their proof is a new method


  • P. Bourgade
  • Mathematics
    Proceedings of the International Congress of Mathematicians (ICM 2018)
  • 2019
We survey recent mathematical results about the spectrum of random band matrices. We start by exposing the Erd{\H o}s-Schlein-Yau dynamic approach, its application to Wigner matrices, and extension



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

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.

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.