2012/07/31 by Antti Knowles, Jun Yin · 1 citation
Mathematics · Physics and Astronomy · #math.PR #math-ph #math.MP
paper · pdf · doi:10.1214/13-aop855
published as Annals of Probability 2014, Vol. 42, No. 5, 1980-2031 · Published in at http://dx.doi.org/10.1214/13-AOP855 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
arxiv created 2014/09/03 · arxiv updated 2014/09/04
We derive the joint asymptotic distribution of the outlier eigenvalues of an additively deformed Wigner matrix H. Our only assumptions on the deformation are that its rank be fixed and its norm bounded. Our results extend those of [The isotropic semicircle law and deformation of Wigner matrices. Preprint] by admitting overlapping outliers and by computing the joint distribution of all outliers. In particular, we give a complete description of the failure of universality first observed in [Ann. Probab. 37 (2009) 1-47; Ann. Inst. Henri Poincaré Probab. Stat. 48 (1013) 107-133; Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices. Preprint]. We also show that, under suitable conditions, outliers may be strongly correlated even if they are far from each other. Our proof relies on the isotropic local semicircle law established in [The isotropic semicircle law and deformation of Wigner matrices. Preprint]. The main technical achievement of the current paper is the joint asymptotics of an arbitrary finite family of random variables of the form \langlev,(H-z)-1w⟩.