# 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…

## 19 Citations

### Universality for 1d Random Band Matrices

- Computer ScienceCommunications in Mathematical Physics
- 2021

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

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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

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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

- Mathematics
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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

- MathematicsThe Annals of Probability
- 2019

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

- MathematicsAnnales Henri Poincaré
- 2019

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

- Mathematics
- 2021

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 $W\ll N^{1/4}$

- Mathematics
- 2022

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

- Mathematics, Physics
- 2019

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…

### RANDOM BAND MATRICES

- MathematicsProceedings 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…

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