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Bulk universality for generalized Wigner matrices

# Bulk universality for generalized Wigner matrices,Laszlo Erdos,Horng-Tzer Yau,Jun Yin

Bulk universality for generalized Wigner matrices
Consider $N\times N$ Hermitian or symmetric random matrices $H$ where the distribution of the $(i,j)$ matrix element is given by a probability measure $\nu_{ij}$ with a subexponential decay. Let $\sigma_{ij}^2$ be the variance for the probability measure $\nu_{ij}$ with the normalization property that $\sum_{i} \sigma^2_{ij} = 1$ for all $j$. Under essentially the only condition that $c\le N \sigma_{ij}^2 \le c^{-1}$ for some constant $c>0$, we prove that, in the limit $N \to \infty$, the eigenvalue spacing statistics of $H$ in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth $M$ the local semicircle law holds to the energy scale $M^{-1}$.
Published in 2010.
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## Citation Context (1)

• ...In [7], Erd˝ os, Yau, and Yin used different methods to prove an analogue of (6) for finite band matrices of size N × N ,f or� W −1 log C N , and with error term (W �) −1/2 log C N ...

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## Citations (9)

### The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices(Citations: 1)

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### Myocardial Ischemia and Ventricular Tachycardia on Continuous Electrocardiographic Monitoring and Risk of Cardiovascular Outcomes After Non–ST-Segment Elevation Acute Coronary Syndrome (from the MERLIN-TIMI 36 Trial)

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### Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model(Citations: 4)

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