CLT for spectra of submatrices of Wigner random matrices

Citation Borodin, Alexei. "CLT for spectra of submatrices of Wigner random matrices. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Abstract. We prove a CLT for spectra of submatrices of real symmetric and Her-mitian Wigner matrices. We show that if in the standard normalization the fourth moment of the off-digonal entries is GOE/GUE-like then the limiting Gaussian process can be viewed as a collection of simply yet nontrivially correlated two-dimensional Gaussian Free Fields. Introduction. Gaussian global fluctuations of eigenvalues of GUE, GOE, Wigner random matrices, and their generalizations is a well-studied subject, see e.g. Chapter 2 of [AGZ] and Chapter 9 of [BS] as well as references therein. One would usually concentrate on studying the spectrum of the full matrix, but it comes as no surprise that for large submatrices with a regular limiting behavior, the joint fluctuations would still be Gaussian. We prove this fact by a slight modification of the moment method presented in [AGZ]. It becomes more interesting when one looks at the limiting covariance structure. In what follows we assume that in the standard normalization the fourth moment of the off-diagonal entries of our matrices is the same as for GOE/GUE. The first statement is that for such a (real symmetric or Hermitian) Wigner matrix, the joint fluctuations of spectra of nested submatrices formed by cutting out top left corners are described by the two-dimensional Gaussian Free Field (GFF), see e.g. [S] for definitions and basic properties of GFFs. Although this result seems to be new, the appearance of the GFF is also not too surprising. Indeed, as was shown in [JN] and [OR], for GUE the eigenvalue ensemble of nested matrices arises as a limit of random surfaces, and for random surfaces the relevance of the GFF is widely anticipated, see [K], [BF] for rigorous results and further references. One might argue however that the GFF interpretation simplifies the description of the covariance in the one-matrix case, cf. Proposition 3 below. The real novelty comes when one considers joint fluctuations for different nested sequences of submatrices. For each of the nested sequences the fluctuations are again described by the GFF. On the other hand, when different sequences have nontrivial and asymptotically regular intersections, these GFFs are correlated, and the exact form of the covariance kernel turns out to be simple. One could argue that …