Random matrices: Universality of local spectral statistics of non-Hermitian matrices
By Terence Tao, Van H. Vu. 2012
182 citations recorded by the source corpus.
It is a classical result of Ginibre that the normalized bulk k-point correlation functions of a complex n×n Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on C with kernel K∞(z,w):=1πe−|z|2/2−|w|2/2+zw¯ in the limit n→∞. In this paper, we show that this asymptotic law is universal among all random n×n matrices Mn whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts and whose moments match that of the complex Gaussian ensemble to fourth order. Analogous results at the edge of the spectrum are also obtained. As an application, we extend a central limit theorem for the number of eigenvalues of complex Gaussian matrices in a small disk to these more general ensembles. These results are non-Hermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a key new difficulty arises in the non-Hermitian case, due to the instability of the spectrum for such matrices. To resolve this issue, we the need to work with the log-determinants log|det(Mn−z0)| rather than with the Stieltjes transform 1ntr(Mn−z0)−1, in order to exploit Girko’s Hermitization method. Our main tools are a four moment theorem for these log-determinants, together with a strong concentration result for the log-determinants in the Gaussian case. The latter is established by studying the solutions of a certain nonlinear stochastic difference equation. With some extra consideration, we can extend our arguments to the real case, proving universality for correlation functions of real matrices which match the real Gaussian ensemble to the fourth order. As an application, we show that a real n×n matrix whose entries are jointly independent, exponentially decaying and whose moments match the real Gaussian ensemble to fourth order has 2nπ−−√+o(n√) real eigenvalues asymptotically almost surely.
Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/b5ab60b80e19c449f7815993842fddeebfca2451
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In Discrete Mathematics and Combinatorics
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Citations in the corpus
- Local Universality of Zeroes of Random Polynomials (2013, methodology): “logn in this case (see Wang [1983], Edelman & Kostlan [1996] for more precise asymptotics).”
- Online Convex Optimization with Stochastic Constraints (2017, methodology): “Similar proof techniques are previously used in [26, 24] to prove different concentration inequalities for supermartingales/martingales with unbounded differences.”